Districting Strategy Game

ABSTRACT

Games of the present technology are strategic districting games in which one or more players are presented with a region divided into sectors, where each sector has a given set of elements, and players are tasked with combining the sectors to create a plurality of districts that achieve an objective.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser.No. 62/785,824, filed on Dec. 28, 2018, the entirety of which isincorporated by reference herein.

FIELD OF THE INVENTION

The present technology relates to the field of games, and moreparticularly to strategy games.

COPYRIGHT NOTICE

A portion of the disclosure of this patent document contains materialwhich is subject to copyright protection. The copyright owner has noobjection to the facsimile reproduction by anyone of the patentdisclosure, as it appears in the Patent and Trademark Office patentfiles or records, but otherwise reserves all copyright rightswhatsoever.

SUMMARY

Strategy games of the present technology may be played by at least oneplayer. The playing surface comprises a region that includes a boundedshape having an area divided into a plurality of sectors. Each sectorcomprises a bounded shape having an area within the region that does notoverlap with any other sector. Each sector contains a set of elements,each element of the set of elements having a type and quantity. Duringplay, each player makes one move per turn, according to a set of rulesdefining types of moves that can be made by the at least one player andrestrictions governing how districts can be formed from the plurality ofsectors, in pursuit of combining the plurality of sectors into a givennumber of districts in a manner that seeks to achieve a pre-defined goalbased on an aggregation of the elements within each district.

Methods of playing strategy games of the present technology includeproviding a playing surface that includes a region comprising a boundedshape having an area divided into a plurality of sectors. Each sectorcomprises a bounded shape having an area within the region that does notoverlap with any other sector, and each sector contains a set ofelements, each element of the set of elements having a type andquantity. Methods of playing strategy games of the present technologyfurther include making one move per turn per player, according to a setof rules defining types of moves that can be made by the at least oneplayer and restrictions governing how districts can be formed from theplurality of sectors, in pursuit of combining the plurality of sectorsinto a given number of districts in a manner that seeks to achieve apre-defined goal based on an aggregation of the elements within eachdistrict.

BRIEF DESCRIPTION OF THE DRAWINGS

Specific examples have been chosen for purposes of illustration anddescription, and are shown in the accompanying drawings, forming a partof the specification. Within the Figures, like parts have been givenlike numbers for ease of reference. It should be understood that thedrawings are not necessarily drawn to scale and that they are intendedto be merely illustrative.

FIG. 1 illustrates a first example of a strategy game of the presenttechnology, having a first region.

FIG. 2 illustrates an example of a region that can be included in asecond strategy game of the present technology.

FIG. 3 illustrates examples of possible district shapes having foursectors that can be formed during play of a strategy game using theregion of FIG. 2.

FIG. 4 illustrates a possible solution of a strategy game using theregion of FIG. 2 in accordance with a first pre-defined goal.

FIG. 5 illustrates a possible solution of a strategy game using theregion of FIG. 2 in accordance with a second pre-defined goal.

FIG. 6 illustrates a possible solution of a strategy game using theregion of FIG. 2 in accordance with a third pre-defined goal.

FIG. 7 illustrates examples of possible district shapes having sixsectors that can be formed during play of a strategy game using theregion of FIG. 1.

FIG. 8 illustrates a possible solution of a strategy game using theregion of FIG. 1 in accordance with a first pre-defined goal.

FIG. 9 illustrates a possible solution of a strategy game using theregion of FIG. 1 in accordance with a second pre-defined goal.

FIG. 10 illustrates a possible solution of a strategy game using theregion of FIG. 1 in accordance with a third pre-defined goal.

FIG. 11 illustrates an example of a region that can be included in athird strategy game of the present technology.

FIG. 12 illustrates a strategy game including the region of FIG. 11.

FIG. 13 illustrates the scoreboard of FIG. 12.

FIG. 14 illustrates one example of sector tiles that can be used in astrategy game of FIG. 12.

FIG. 15A illustrates a possible arrangement of markers on a regionduring play.

FIG. 15B illustrates a second possible arrangement of markers on aregion during play.

FIG. 16A illustrates a third possible arrangement of markers on a regionduring play.

FIG. 16B illustrates a fourth possible arrangement of markers on aregion during play.

FIG. 17A illustrates a fifth possible arrangement of markers on a regionduring play.

FIG. 17B illustrates a sixth possible arrangement of markers on a regionduring play.

FIG. 18 illustrates examples of sector relationships.

FIG. 19 illustrates a seventh possible arrangement of markers on aregion during play.

FIG. 20 illustrates an eighth possible arrangement of markers on aregion during play.

FIG. 21 illustrates one possible position for the game of FIG. 11 after15 turns.

FIG. 22 illustrates one possible position for the game of FIG. 21 after23 turns.

FIG. 23 illustrates one possible position for the game of FIG. 22 after27 turns.

FIG. 24 illustrates one possible position for the game of FIG. 23 after29 turns.

FIG. 25 illustrates one possible position for the game of FIG. 24 after32 turns.

FIG. 26 illustrates one possible position for the game of FIG. 25 after34 turns.

FIG. 27 illustrates one possible position for the game of FIG. 26 after35 turns.

FIG. 28 illustrates one possible position for the game of FIG. 27 after36 turns.

FIG. 29 illustrates one possible position for the game of FIG. 28 after37 turns.

FIG. 30 illustrates one possible position for a final position of thegame of FIG. 29.

FIG. 31 illustrates a guide for arrangement of a symmetric game of FIG.11.

FIG. 32 illustrates a symmetric game setup table for a symmetric game ofFIG. 11.

FIG. 33 illustrates a fourth example of a strategy game of the presenttechnology.

FIG. 34 illustrates one possible initial position for a strategy game ofFIG. 33.

FIG. 35 illustrates one possible final position for a strategy game ofFIG. 34.

FIG. 36 illustrates a fifth example of a strategy game of the presenttechnology.

FIG. 37 illustrates a scoreboard that may be used with the strategy gameof FIG. 36.

FIG. 38A illustrates a possible solution by a first player in phase oneof a first strategy game of FIG. 36.

FIG. 38B illustrates a possible solution by a second player in phase oneof the strategy game of FIG. 38A.

FIG. 39A illustrates a possible solution by a first player in phase twoof the strategy game of FIG. 38A.

FIG. 39B illustrates a possible solution by a second player in phase twoof the strategy game of FIG. 38A.

FIG. 40A illustrates a possible solution by a first player in a secondstrategy game of FIG. 36.

FIG. 40B illustrates a possible solution by a second player in thestrategy game of FIG. 40A.

FIG. 41 illustrates examples of sector tiles in a strategy game of thepresent technology having four elements.

FIG. 42A illustrates an initial position in a region of a sixth exampleof a strategy game of the present technology.

FIG. 42B illustrates one possible final position for a strategy game ofFIG. 42A.

FIG. 43 illustrates a scoreboard that may be used in the strategy gameof FIG. 42A.

FIG. 44 illustrates a seventh example of a strategy game of the presenttechnology.

FIG. 45 illustrates one possible initial position for a strategy game ofFIG. 44.

FIG. 46 illustrates one possible final position for a strategy game ofFIG. 45.

FIG. 47 illustrates one possible final position for an eighth example ofa strategy game of the present technology.

FIG. 48 illustrates a scoreboard that may be used in a strategy game ofFIG. 47.

FIG. 49 illustrates a game board that may be used with a ninth exampleof a strategy game of the present technology.

FIG. 50 illustrates one possible initial position for a tenth example ofa strategy game of the present technology.

FIG. 51 illustrates examples of sector tiles that may be used in astrategy game of FIG. 50.

FIG. 52 illustrates a scoreboard that may be used in a strategy game ofFIG. 50.

FIG. 53 illustrates a rotationally symmetric sector arrangement for astrategy game of FIG. 50.

FIG. 54 illustrates one possible region that may be used in an eleventhexample of a strategy game of the present technology.

FIG. 55 illustrates examples of sector tiles that may be used in astrategy game of FIG. 54.

FIG. 56 illustrates one possible initial position in a region that maybe used in a twelfth example of a strategy game of the presenttechnology.

FIG. 57 illustrates part 1 of scoreboard that may be used in thestrategy game of FIG. 56.

FIG. 58 illustrates part 2 of scoreboard that may be used in thestrategy game of FIG. 56.

FIG. 59 illustrates one possible position for the game of FIG. 56 duringplay.

FIG. 60 illustrates one possible position for the game of FIG. 59 after40 turns.

FIG. 61 illustrates one possible position for the game of FIG. 60 after41 turns.

FIG. 62 illustrates one possible position for the game of FIG. 61 after42 turns.

FIG. 63 illustrates one possible position for the game of FIG. 62 after65 turns.

FIG. 64 illustrates one possible position for the game of FIG. 63 after70 turns.

FIG. 65 illustrates one possible position for the game of FIG. 64 after75 turns.

FIG. 66 illustrates one possible position for the game of FIG. 65 after79 turns.

FIG. 67 illustrates one possible position for the game of FIG. 66 after81 turns.

FIG. 68 illustrates one possible position for the game of FIG. 67 after83 turns.

FIG. 69 illustrates one possible position for the game of FIG. 68 after84 turns.

FIG. 70 illustrates one possible final position for phase 1 the game ofFIG. 69.

FIG. 71 illustrates one possible initial position in a region that maybe used in a thirteenth example of a strategy game of the presenttechnology.

FIG. 72 illustrates part 1 of scoreboard that may be used in thestrategy game of FIG. 71.

FIG. 73 illustrates part 2 of scoreboard that may be used in thestrategy game of FIG. 71.

FIG. 74 illustrates one possible final position for the strategy game ofFIG. 71.

FIG. 75 provides a table that includes a key to symbols used in FIGS.1-74.

DETAILED DESCRIPTION

The present technology provides strategy games involving the division ofa region into districts to achieve one or more predefined goals. The useof the prefix “pre” herein means any time prior to beginning the play ofthe game. Games of the present invention can be provided as board games,sector based games, paper based games, electronic games, or games on anyother suitable presentation medium. For example, the playing surface onwhich a region may be provided may take the form of a board, a set ofsectors, a piece of paper, a three-dimensional form, or a screen.

Table 1, provided in FIG. 75, is a key that explains the meaning of thevarious numbers and shapes that appear in the Figures. The first 5 rowsof Table 1 represent designations that may be used with sectors. Thefirst row of Table 1 is a sector number, and each sector within a regionmay have a distinct sector number to distinguish the sector from anyother sector within the region. The fifth row represents the totalpopulation within a sector. The second through fourth rows representdesignations of a party majority, for use in examples of strategy gamesthat include political parties, which are represented in the Table asincluding a red party, a blue party, and a green party. The final row ofTable 1 provides a designation for a scoring token, which may be used inexamples of strategy games that have a scoreboard. The remainder of therows of Table 1 provide representations of designations for markers,including home base markers and expansion markers, that may be placedonto a sector during a player's move to assign that sector to adistrict.

FIG. 1 illustrates one example of a strategy game 100 of the presenttechnology. The strategy game 100 includes a region 102, which is ahexagon. Generally, in strategy games of the present technology, theregion has at least one area or volume bounded by pre-defined externalboundaries. The region can be any real or imagined boundedtwo-dimensional shape or three-dimensional form, such as a polygon orother geometric shape, a polyhedron, or a geographical area. In manyexamples, the region is presented as a map. While the region ispreferably defined by a single bounded shape or form, a region caninclude multiple bounded shapes or forms. In examples where a region hasmultiple bounded shapes or forms, it is preferred that at least aportion of each bounded shape or form be predefined to be connected withat least part of one of the other bounded shapes or forms.

The region 102 is pre-divided into a plurality of sectors 104. Thesectors are stationary, and do not move during play of the game. Eachsector has an area or volume within the region bounded by pre-definedboundaries. Each sector is distinct and does not overlap with any othersector. Each sector can be any real or imagined bounded shape orform—such as a polygon or other geometric shape, a polyhedron or otherthree-dimensional form, or a geographical area—within the region. Forexample, each sector 104 is a triangle. For purposes of the strategygames described herein, a sector cannot be further divided. Every partof the region 102 is defined as being part of a sector, and,collectively, the sectors cover the entire area or volume of the region102.

Each sector 104 contains a set of elements, each set of elementsincluding one or more elements. The set of elements in a sector is acomplete list of the elements contained in the sector, and each elementof the set of elements has a type and quantity. The quantity of anelement may be any amount, and is preferably greater than zero. In someexamples, the type of each element is voters that favor a particularpolitical party. In other examples, the type of each element may be aresource (i.e., something useful), a hazard (i.e., something harmful),or scrap (i.e., something neither useful nor harmful). Each type ofelement may have the same value to each player. Alternatively, eachelement may have a value that is player-specific. That is, the sameelement may be a resource (i.e., something useful) to one player but ahazard (i.e., something harmful) or scrap (i.e., something neitheruseful nor harmful) to another player.

Referring to FIG. 1, the illustrated strategy game 100 is a game inwhich the type of element in each sector is voters that favor aparticular political party, and the quantity is the margin of voterswithin the sector that favor the indicated political party. For example,voters in sector 106 favor the Blue party by a margin of 2, and votersin sector 108 favor the Red party by a margin of 9. The voter margin mayrepresent single voters, hundreds of voters, thousands of voters, or anyother suitable amount of voters, as appropriate for a given game.

The strategy game 100 shown in FIG. 1 also includes a set of markers110, wherein the set of markers comprises a plurality of marker subsets112, each marker subset representing a district. Specifically, eachmarker within each marker subset 112 is configured to be placed on asector to assign the sector as being part of the district represented bythe marker subset 112. Each marker subset 112 comprises a plurality ofmarkers, and may include a home base marker 114 and at least one,preferably more than one, expansion marker 116. In strategy game 100,there are fifty-four sectors 104, and nine marker subsets 112, eachmarker subset 112 representing one of nine districts will be createdduring the game. In order for each sector 104 to have a marker on it atthe end of the game, there are preferably at least six markers in eachmarker subset 112. In this example, there may be one home base marker114 and at least five expansion markers 116 in each marker subset 112.

The strategy game 100 shown in FIG. 1 also includes a scoreboard 118.The scoreboard 118 has one row for each of the nine districts that willbe created during the game, each row containing a scoring spectrum, suchas the illustrated spectrum of which party (Red or Blue) is favored andthe margin by which the party is favored within the district. Each rowhas a scoring token 120 that may be used keep track of the favored partyand total voter margin within each district.

Games of the present technology may be played by at least one player.Some embodiments are designed to be played by a single player, whileother embodiments are designed to be played by a plurality of players,such as at least two players. The term “player” as used herein can meanone individual, or a team of individuals. In many games of the presenttechnology, two or more players take alternating turns, and each playermust make one move per turn. In some embodiments of the presenttechnology there is only one player who faces an individual challenge.In some games of the present technology, two or more playersindependently consider the exact same challenge, each using a separatecopy of identical game components. In such examples, each player maytake turns independently of the other players, using their own set ofthe game components. Turns may or may not be under time constraints. Theplayer who does the best job of attaining the pre-defined goal wins thegame.

Generally, in order to play a game of the present technology, one ormore players are provided with a region that has been divided intonon-overlapping sectors that (i) may not be further divided, (ii) do notoverlap, and (iii) together cover the area or volume of the largeregion. At the outset, the one or more players are informed of eachsector's precise shape, location, and set of elements, and suchinformation may be depicted graphically. In most examples, there are nodistricts yet created within the region at the start of play, and eachsector is initially considered to be unassigned.

During play, each player makes one move per turn by assigning a sectorto a district, according to a set of rules defining types of moves thatcan be made by the at least one player and restrictions governing howdistricts can be formed from the plurality of sectors, in pursuit ofcombining the plurality of sectors into a given number of districts in amanner that seeks to achieve a pre-defined goal based on an aggregationof the elements within each district. Examples of pre-defined goalsinclude, but are not limited to: (a) maximizing a portion of the givennumber of districts that contain at least a certain level of at leastone of the elements of the set of elements; (b) minimizing a portion ofthe given number of districts that contain at least a certain level ofat least one of the elements of the set of elements; and (c) maximizinga number of points earned by at least one of the players, where thenumber of points earned by the at least one player depends on theaggregation of the elements within each district. In examples ofstrategy games of the present technology that include at least twopolitical parties, such as strategy game 100 of FIG. 1, examples ofpre-defined goals include, but are not limited to: (a) maximizing aportion of the given number of districts controlled by one of thepolitical parties, and (b) equalizing a portion of the given number ofdistricts controlled by each of the political parties.

Rules of strategy games of the present technology define types of movesthat can be made by the at least one player and restrictions governinghow districts can be formed from the plurality of sectors. For example,the rules may provide at least two general categories of moves that aplayer can make:

-   -   1. Establish a new district by assigning the first sector that        belongs to it.    -   2. Expand a district by assigning an unassigned sector to an        already established district.

In some examples, the rules include additional categories of moves thatare permitted. For example, three additional categories of moves couldbe:

-   -   3. Reassign a sector from one district to another district.    -   4. Break up one or more adjacent districts by un-assigning all        sectors that belong to them.    -   5. Freeze a district. No player may modify the district during        the next several turns.

As another example, rules regarding how districts may be formed mayinclude:

-   -   A. Every sector within the region must be assigned to exactly        one district.    -   B. Each district must be a single connected piece.    -   C. Each district may be required to have at least, or at most, a        certain number of sectors, or may be required to include a        specific pre-defined number of sectors.

In many games of the present technology, play concludes when there areno more permitted moves, or when every sector within the region has beenassigned to a district. If there are two or more players, the winningplayer is the player that does the best job of achieving the pre-definedgoal. If there is one player, the player wins if the pre-defined goal isachieved, and otherwise loses.

Computer games incorporating the concept may be played in at least twomodes. In mode 1, a computer plays the role of (i.e., makes thedecisions for) one or more players. In this mode, the computer makes useof sophisticated artificial intelligence techniques that are programmedinto it ahead of time by a team of expert computer scientists. In mode2, the computer provides visualization, data storage, and communicationservices to facilitate play but does not participate as a decision makerduring play.

Taxonomy of Strategy Games of the Present Technology:

Strategy games of present technology include hundreds of recreationaland non-recreational games and puzzles. A taxonomy for the exemplarystrategy games is provided below, and is based on a nine-part code. Thiscode summarizes the main aspects of a given example of the presenttechnology. Each part of the code contains one or more capital lettersor integers and is separated from the other parts of the code by forwardslashes. The generic code for an example of the current technology is asfollows:

Part1/Part2/Part3/Part4/Part5/Part6/Part7/Part8/Part9

Part 1 of the code is either the letter “A” or “D.” It is “A” if theexample is analog in nature; it is “D” if the example is digital innature.

Part 2 is either “Z” or “G.” It is “Z” if the example is a single-playerpuzzle (e.g., an individual challenge like a Sudoku puzzle). It is “G”if the example is a multi-player game.

Part 3 refers to the shape of each sector and the number of sectors. Itconsists of a letter followed by an integer with no intercedingpunctuation. It begins with “S” if each sector is a square; “T” if eachsector is an equilateral triangle; “H” if each sector is a regularhexagon; “C” if each sector is a complex, two-dimensional shape such asthe shape of a real-world county or country; and “0” if each sector isanother shape (e.g., a three-dimensional form). The integer Y thatfollows the letter indicates how many sectors are in the game. If theletter is (S, T, H), the number of sectors is (36*Y, 54*Y, 37*Y)respectively or slightly less than this. If the letter is “C” or “0,”the value Y gives the exact number of sectors.

Part 4 is either “P” or “N.” It is “P” if the example focuses onpolitics. It is “N” if the example does not focus on politics.

Part 5 is an integer that gives the number of element types that arefound within the sectors. Its value often ranges from 2-6.

Part 6 is either the letter “V” or an integer. If it is the letter “V,”the number of districts to be formed is variable and is unknown at thestart. Otherwise, the number of districts to be formed is known at thestart and equals the value in this part of the code.

Part 7 specifies the game paradigm. It only applies to multi-playergames, thus it exists only if part 2 of the code is “G.” Part 7 is “U”if the game involves alternating, turn-based play. It is “I” if the gameinvolves simultaneous independent play in which each player takes turnsindependently of the others. Games with simultaneous independent playcan have any number of players, whereas games with alternating,turn-based play typically have no more than 6 players.

Parts 8 and 9 only apply to multi-player games with alternating,turn-based play; these parts of the code exist only if part 2 of thecode is “G” and part 7 of the code is “U.”

Part 8 indicates the number of players in the game. It is expressed as arange—with two integers separated by a hyphen—if different numbers ofplayers can play the game. It is a single integer if the game isdesigned for a specific number of players (e.g. for two players only).

Part 9 contains one or more of the letters “E,” “X,” “R,” “B,” and “F.”These five letters respectively refer to five categories of allowedmoves—“Establish,” “Expand,” “Reassign,” “Break up,” and “Freeze”—whichare briefly described in a previous paragraph. This part of the codecontains the letters that correspond to the categories of moves that areallowed in the game.

The description of each example provided below begins with a discussionof its taxonomic code. This code gives the reader a quick understandingof the example's main aspects. One or more parts of a code may containthe question mark symbol “?” if those aspects are unspecified.

EXAMPLES

Several non-limiting examples of strategy games of the presenttechnology are provided below. While the examples use numbers, letters,and generic shapes to distinguish between different districts, elementtypes, and sectors, it should be noted that other methods of distinctioncould be used. For example, colors or specialized graphics could beused.

Example 1

Strategy games of the Example 1 have a taxonomic code A/Z/S1/P/2/9. Theyare analog, single-player puzzles with square sectors and a politicalfocus in which two types of elements are present in the sectors and ninedistricts are formed. A nearly unlimited number of possible instances ofthis kind of puzzle can be created, one of which is illustrated in FIG.2 as strategy game 200. A collection of instances of this kind ofstrategy game can be assembled in a booklet.

Two aspects distinguish this kind of puzzle from most other types oflogic puzzles. First, there may be multiple solutions to a given puzzle;a unique solution is not guaranteed. Second, the “partial solution”concept does not apply. In other words, if one partially finishes apuzzle, there is no guarantee that the partial solution will give riseto a complete solution. A logical “guess and check” approach isrecommended for solving this kind of puzzle.

In strategy game 200, the player is given a map of a square shapedregion 202 that has been divided into 36 square shaped sectors 204,which are arranged in six rows and six columns. The player is taskedwith dividing the region 200 into a given number of politicaldistricts—i.e. to draw lines that define the boundaries of thedistricts—in order to achieve the stated objective.

Two types of elements—two political parties—occupy the region. Oneelement is the Red Party and the other element is the Blue Party. Eachsector 204 may represent a community, and each community has a numberwhich is the community's voter margin. A black number in a white circlemeans that there are more Red Party supporters than Blue Partysupporters in the community (see the key in Table 1). In such a case,the community favors the Red Party. A white number in a black circlemeans that there are more Blue Party supporters than Red Partysupporters in the community. In such a case, the community favors theBlue Party. The number itself is the margin (in thousands of voters) bywhich the community supports one party over the other. For example,sector 206 has a black 3 in a white circle, which may mean that thereare 3000 more Red Party supporters than Blue Party supporters in thatcommunity. In this case, the community's voter margin is “+3 Red.” Insector 208, there is a white 4 in a black circle, which may mean thatthere are 4000 more Blue Party supporters than Red Party supporters inthat community. The voter margin in sector 208 is “+4 Blue.” In Sector210, there is a zero, which means that the community equally supportsthe two parties. In such a case, the community's voter margin is 0.

In each puzzle of Example 1, the player is asked to divide the regioninto 9 political districts. In other words, the player is asked to drawlines that define the boundaries of 9 political districts.

The rules for forming political districts are as follows.

-   -   1. Each sector must belong in its entirety to one and only one        district.    -   2. No two districts may overlap.    -   3. Each district must consist of four sectors that form a single        connected piece.

FIG. 3 shows examples of shapes 302, 304, 306, 308, and 310, in which adistrict may be formed. Each shape of FIG. 3 is formed from four sectors204 that form a single connected piece. Rotations and reflections ofshapes 302-310 would also be acceptable.

When a district is formed, the player must pay attention to its votermargin (i.e. margin). A district's voter margin indicates which partyhas more voters in the district. A district's voter margin depends onthe voter margins of the sectors in the district. It equals thedifference between the sum of the black and white numbers in thedistrict. The voter margin favors the Red Party if the sum of the blacknumbers in white circles exceeds the sum of the white numbers in blackcircles; it favors the Blue Party if the opposite is true; and it iszero if the sum of the black numbers equals the sum of the white numbersin a district.

For example, if a district has four sectors with voter margins “+4 Red,”“+6 Blue,” “0,” and “+7 Red,” then the district's voter margin is “+5Red” (=4+7+0−6). In other words, there are 5000 more Red Partysupporters than Blue Party supporters in the district. The party withmore voters in a district is said to control the district. Neither partycontrols a district—a district is tied—if the district's voter margin is0.

In each puzzle, the player may be asked to pursue one of three goals:

-   -   A. Create political districts that maximize the advantage of the        Red Party    -   B. Create political districts that maximize the advantage of the        Blue Party    -   C. Create political districts that equalize the advantage of the        two parties

Goals A, B, and C relate to the voter margins of the districts that areformed. In exact terms, Goal A is to, first and foremost, maximize thenumber of districts controlled by the Red Party and, secondarily,maximize the margin by which the Red Party controls its least safedistrict. Goal B is to do the same except to the benefit of the BlueParty. Goal C is to (i) equalize the number of districts controlled byeach party, (ii) equalize the margin by which each party controls itsleast safe district, and (iii) maximize the number of tied districtsthat have a voter margin of 0. A party's least safe district is thedistrict in which it has the smallest majority.

Each puzzle may have an easy version and a hard version. The easyversion asks the player to pursue the goal at hand—A, B, or C—to amodest extent. The hard version asks the player to pursue the same goalto the maximum possible extent.

There are different solutions to the strategy game 200 for each goal A,B, and C. For example, FIG. 4 shows a solution in which the region 202has been divided into nine districts 212-228 to achieve goal A. In thesolution of FIG. 4, the Red party controls seven of the districts by amargin of at least +2. These include districts 212, 214, 218, 220, 222,224, and 228. In FIG. 5, the region 202 has been divided into ninedistricts 230-246 to achieve goal B. In the solution of FIG. 5, the Blueparty controls seven of the districts by a margin of at least +2. Theseinclude districts 230, 232, 234, 240, 242, 244, and 246. In FIG. 6, theregion 200 has been divided into nine districts 248-264 to achieve goalC. In the solution of FIG. 6, each party controls two districts, fivedistricts are tied, and the margin by which each party controls itsleast safe district is +9. Districts 248, 250, 252, 254, and 262 aretied. Districts 256 and 260 are controlled by the Red Party and havevoter margins of “+9 Red” and “+12 Red” respectively. Districts 258 and264 are controlled by the Blue Party and have voter margins of “+12Blue” and “+9 Blue” respectively.

Example 2

Strategy games of the Example 2 have a taxonomic code A/Z/T1/P/2/9. Theyare analog, single-player puzzles with triangular sectors and apolitical focus in which two types of elements are present in thesectors and nine districts are formed.

In some examples of this kind of puzzle, the region may be a hexagonalregion that has 54 triangular sectors (i.e. communities). Each communitymay have the same population. An example of such a region is region 102in FIG. 1.

In the example shown in FIG. 1, supporters of two political parties—Redand Blue—occupy the region 102. Each sector 104 represents a community,and has a number which represents the community's voter margin. A blacknumber in a white circle means that there are more Red Party supportersthan Blue Party supporters in the community. A white number in a blackcircle means that there are more Blue Party supporters than Red Partysupporters in the community. The number within each sector is the margin(in thousands of voters) by which the community supports one party overthe other.

In each puzzle of Example 2, the player is asked to divide the regioninto 9 political districts. In other words, the player is asked to drawlines that define the boundaries of 9 political districts.

The rules for forming political districts are as follows.

-   -   1. Each community must belong in its entirety to one and only        one district.    -   2. No two districts may overlap.    -   3. Each district must consist of six sectors that form a single        connected piece.

FIG. 7 shows examples of shapes 402-424 in which a district may beformed. Each shape of FIG. 7 is formed from six sectors 104 that form asingle connected piece. Rotations and reflections of shapes 402-424would also be acceptable.

When a district is formed, the player must pay attention to its votermargin. Just as in Example 1, a district's voter margin equals thedifference between the sum of the black and white numbers in thedistrict. The voter margin favors the Red Party if the sum of the blacknumbers in white circles exceeds the sum of the white numbers in blackcircles; it favors the Blue Party if the opposite is true; and it iszero if the sum of the black numbers equals the sum of the white numbersin a district.

In each puzzle, the player may be asked to pursue one of three goals:

-   -   A. Create political districts that maximize the advantage of the        Red Party    -   B. Create political districts that maximize the advantage of the        Blue Party    -   C. Create political districts that equalize the advantage of the        two parties

Each puzzle may have an easy version and a hard version. The easyversion asks the player to pursue the goal at hand—A, B, or C—to amodest extent. The hard version asks the player to pursue the same goalto the maximum possible extent.

There are different solutions to the strategy game 100 for each goal A,B, and C. For example, FIG. 8 shows a solution in which the region 102has been divided into nine districts 122-138 to achieve goal A. In thesolution of FIG. 8, the Red party controls eight of the districts—allexcept district 134—by a margin of at least +1. In FIG. 9, the region102 has been divided into nine districts 140-156 to achieve goal B. Inthe solution of FIG. 9, the Blue party controls eight of thedistricts—all except district 146—by a margin of at least +1. In FIG.10, the region 102 has been divided into nine districts 158-174 toachieve goal C. In the solution of FIG. 10, each party controls threedistricts, three districts are tied, and the margin by which each partycontrols its least safe district is +2. Districts 162, 172, and 174 aretied. Districts 160, 164, and 166 are controlled by the Red Party andhave voter margins of “+2 Red,” “+6 Red,” and “+17 Red” respectively.Districts 158, 168, and 170 are controlled by the Blue Party and havevoter margins of “+12 Blue,” “+2 Blue,” and “+11 Blue” respectively.

Example 3

Strategy games of the Example 3 have a taxonomic codeA/G/S1/P/2/9/U/2/EXR. They are analog, multi-player games with squaresectors and a political focus in which two types of elements—namely twopolitical parties—are present and nine districts are formed. The gameproceeds according to alternating, turn-based play; there are twoplayers; and moves in categories “E,” “X,” and “R” are allowed.

FIGS. 11-32 illustrate a strategy game 500, having a region 502. Instrategy game 500, two players representing opposing politicalparties—Red and Blue—vie for political control of a square state (region502) by competitively creating nine political districts out of 36 squaresectors 504 in alternating, turn-based fashion. FIG. 11 shows theinitial arrangement of the region 502, prior to being divided intodistricts by the game play. FIG. 30 shows one possible final arrangementof the region 502, after the region 502 is divided into nine districtsby game play.

In strategy game 500, each sector represents a community. The regionrepresents a state and has an American-style, two-party political systemin which one person is elected to represent each political district. Atthe start, the districts have not been formed and the players know thelocation and political composition of each sector (i.e., which party itscitizens favor and by how much). During the first phase of the game,players build the political districts by assigning sectors to politicaldistricts one sector at a time, in alternating turns. They may alsoreassign sectors from large districts to adjacent smaller districts inorder to better equalize the district sizes. During the optional secondphase of the game, the voter margin in each district is converted into anumerical likelihood of each party winning the district, and an electionis simulated, which may be done by rolling dice. The winner is theplayer whose party controls more districts than his/her opponent. A tieis possible if players skip phase 2 of the game.

In the final position, shown in FIG. 30, home base markers 600 andexpansion markers 602, of the types shown in Table 1, are shown thathave been placed by the players during the game. Bold lines indicateboundaries between the nine districts 510-526.

The table below shows the final result of this game. The Blue Party winsthis example game by a score of 5 districts to 3 districts (with onetied district):

Party in By How District Control Much Brown Neither 0 (=6 + 3 − 5 − 4)Red Blue 7 (=8 + 6 + 0 − 7) Orange Blue 3 (=5 + 3 − 3 − 2) Yellow Red 8(=9 + 5 − 2 − 4) Green Red 5 (=7 + 6 − 8) Blue Blue 2 (=9 + 2 − 1 − 8)Purple Red 9 (=1 + 2 + 8 − 1 − 1) Pink Blue 1 (=7 + 4 − 6 − 4) Gray Blue9 (=0 + 5 + 7 − 3)

Game Components

In strategy games of this example, the region 502 is laid out andincludes boundaries for the sectors. To allow multiple varied games tobe played, however, the elements of the sectors are not pre-printed onthe region. Instead, sector tiles 610 (FIG. 14) are provided, which maybe shuffled or otherwise reorganized, and laid onto the region at thestart of a game. The set of elements 616 for each sector tile 610 isprovided on the sector tile. FIGS. 12-14 illustrate game components thatmay be used in strategy game 500:

-   -   A region 502, configured to receive sector tiles 610. In this        example, the region is a 6×6 square, forming a 36 square grid,        each square of the grid being numbered for reference from 1 to        36, and being configured to receive a sector tile 610.    -   36 square sector tiles 610, examples of two types of which are        shown in FIG. 14 as sector tiles 612 and 614. The set of        elements 616 on each sector tile 610 are shown as being marked        twice on each sector tile 610, in different orientations for        visibility from different angles, but it should be understood        that the set of elements on a sector tile 610 may be shown in        any suitable manner, such as a single representation as shown in        FIG. 11. The set of elements that may be used for the 36 sector        tiles 610 for strategy game 500 are listed below, and the        quantity of each sector type is shown in parentheses:

+9 Red (1) +5 Red (2) +1 Red (2) +3 Blue (2) +7 Blue (2) +8 Red (2) +4Red (2) 0 (2) +4 Blue (2) +8 Blue (2) +7 Red (2) +3 Red (2) +1 Blue (2)+5 Blue (2) +9 Blue (1) +6 Red (2) +2 Red (2) +2 Blue (2) +6 Blue (2)

-   -   9 home base markers 600 (one for each marker subset representing        one of nine districts: Brown, Red, Orange, Yellow, Green, Blue,        Purple, Pink, Gray)    -   162 expansion markers 602 (eighteen for each marker subset        representing one of the nine districts: Brown, Red, Orange,        Yellow, Green, Blue, Purple, Pink, and Gray)    -   9 scoring tokens 606 (one for each of the nine districts—Brown,        Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray), each        which may be cube-shaped with faces showing the numbers 10, 20,        30, 40, and 50 or any other suitable shape    -   2 ten-sided dice 618 (1 black and 1 white), each showing the        values 0-9    -   Scoreboard 604 (FIG. 13), which may have nine rows 608, one for        each of the nine districts. Each row 608 may contain a scoring        spectrum that indicates which party (Red or Blue) is favored and        the margin by which the party is favored within the district.

Game Setup

Players may decide (yes or no) if a symmetric game will be played and(yes or no) if phase 2 of the game will be played. The “no-no” option isrecommended for beginners. The “yes-no” option is a game of pure skill,whereas the “no-yes” option maximizes the role of luck in the game.Players then decide who plays Red, who plays Blue, and who takes thefirst turn.

The game components may be laid out at the start of the game is shown inFIG. 12. The sector tiles 610 may be drawn one at a time and placed faceup in a grid square of the region 502 to form a sector 504. The sectortiles 610 may be placed onto the region sequentially on grid squares1-36, or in any other suitable manner. The resulting region 502 with itssectors 504 may look like the region represented in FIG. 11.

Sector Tiles

Each sector tile 610 has a set of elements marked at least once thereonthat show the element type (Red Party or Blue Party being favored) andthe quantity (voter margin). A black number in a white circle, such asfirst element set 506 in FIG. 11, means that there are more Red Partysupporters than Blue Party supporters in the community (see the key inTable 1). In this case, we say the community favors the Red Party. Awhite number in a black circle, such as second element set 508 in FIG.11, means that there are more Blue Party supporters than Red Partysupporters in the community. In this case, we say the community favorsthe Blue Party. The number itself is the margin (which may be inthousands of voters) by which the community supports one party over theother. For example, the black 3 in the white circle of second elementset 508 means that there are 3000 more Red Party supporters than BlueParty supporters in the community. In this case, we say that thecommunity's voter margin is “+3 Red.” The white 7 in the black circle offirst element set 506 means that there are 7000 more Blue Partysupporters than Red Party supporters in the community. In this case, wesay that the community's voter margin is “+7 Blue.” A zero means thatthe community equally supports the two parties. In this case, we saythat the community's voter margin is 0. In some examples, everycommunity is designated as having the same total population.

As discussed above, in this example there are a total of thirty sixsector tiles 610. The seventeen sector tiles favoring the Red party areidentical with respect to their voter margins to the seventeen sectortiles favoring the Blue party, and two sectors have a voter margin of 0.Hence, the overall voter margin in the state is 0; the same number ofvoters support each party statewide.

In this example, the thirty six sector tiles 610—which remain in theirinitial positions as sectors 504 once placed for the game—are used asbuilding blocks to form nine political districts that will cover theregion 502. The nine districts are identified by color: Brown, Red,Orange, Yellow, Green, Blue, Purple, Pink, and Gray. Initially, nosector 504 belongs to any district. During the game, players use coloredmarkers to assign communities to political districts. Each communityeventually belongs to exactly one political district. Since ninedistricts will be created from thirty six sectors 504, at the end of thegame the size of the average district—the number of sectors it has—willbe four. However, the rules may permit variation in the size of adistrict, so some districts may be smaller or larger than others.

The voter margin of a district depends on the voter margins of thesectors 504 that comprise it. The voter margin of a district equals thedifference between the sum of the black and white numbers in thedistrict. The voter margin favors the Red Party if the sum of the blacknumbers in white circles exceeds the sum of the white numbers in blackcircles; it favors the Blue Party if the opposite is true; and it iszero if the sum of the black numbers equals the sum of the white numbersin a district. A district's voter margin indicates which party has morevoters in the district. For example, if a district has four communitieswith voter margins “+4 Red,” “+6 Blue,” “0,” and “+7 Red,” then thedistrict's voter margin is “+5 Red” (=4+7+0−6). In other words, thereare 5000 more Red Party supporters than Blue Party supporters in thedistrict. The party with more voters in a district is said to controlthe district. Neither party controls a district—a district is tied—ifthe district's voter margin is 0.

Scoreboard and Scoring Tokens

During the game, the current voter margin of each district is indicatedby the position and orientation of its scoring token 606 on thescoreboard 604. In particular, each district's scoring token 606 mustalways be placed so that (1) the value in the square it occupies plus(2) the number on the side of the scoring token that faces up equals thedistrict's current voter margin. A scoring token may be cubed shaped,and may have its faces marked in the following manner: unmarked, 10, 20,30, 40, 50. The side of a scoring token 606 that should be face updepends upon the range of the voter margin, and may correspond tounmarked: 0-12, 10:13-22, 20:23-32, 30:33-42, 40:43-52, 50:53-62. Thescoring token 606 for a district may be placed on a square within itsrow 608 when a district's voter margin favors the (Blue, Red) Partyrespectively. For example, consider a moment in the game when threecommunities with voter margins “+5 Blue,” “+1 Red,” and “+4 Blue” havebeen assigned to the Green District. In this case, the Green District'svoter margin is “+8 Blue” (=5+4−1), so the green scoring token should beplaced on square “Green District Voter Margin=+8 Blue” with its unmarkedside facing up. If a community with voter margin “+9 Blue” were added tothis district, its new voter margin would be “+17 Blue” (=9+5+4−1), andthe green scoring token would be moved to square “Green District VoterMargin=+7 Blue” with its “10” side facing up. Alternatively, if acommunity with voter margin “+9 Red” were added to this district, itsnew voter margin would be “+1 Red” (=9+1−5−4), and the green scoringtoken would be moved to square “+1 Red” with its unmarked side facingup.

Playing the Game

Play may include the following three phases, although the second phaseis optional.

-   -   1. Build political districts    -   2. Run an election (optional)    -   3. Identify the winner

Phase 1: Build Political Districts Summary

The first phase is the main phase of the game. During this phase,players may take turns assigning sectors 504 to political districts, onesector at a time, until every sector belongs to a political district.The assignment of a sector 504 to a political district is accomplishedby placing a home base marker 600 or expansion marker 602 on a vacantsector 504. Players may also reassign sectors from large districts toadjacent smaller districts to better equalize the district sizes. Thisis done by changing the color of the marker on a sector 504. At the endof this phase, there will be 9 non-overlapping politicaldistricts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, andGray—that cover the region 502.

Each district evolves in the same way. Initially, it is formless. Atsome point, it is established when its home base marker 600 is placed ona vacant sector 504. (A vacant sector is a sector with no marker on it.)It is then expanded whenever one of its expansion markers 602 is placedon a vacant sector 504 that is adjacent to a sector 504 that alreadybelongs to the district.

The process of building political districts is relatively unrestricted.There is no general requirement for the sequence in which, or locationswhere, districts are constructed. Once begun, the construction of adistrict may be temporarily halted while players take turnsestablishing, expanding, and/or resizing other districts. There is nodistrict size requirement. However, the rules encourage the creation ofdistricts having four sectors 504.

Importantly, all marker subsets 620 (consisting of the home base markerand expansion markers for a given color) and all sectors 504 areavailable to all players. No player “owns” any marker subset or sector504. As long as the rules below are followed, any player may contributeto building any district during any turn. No matter which playerestablished a district, any other player may expand the district orreassign a sector 504 from that district to another district.

Details

Note: During play, the voter margins of all sectors 504 are visible toboth players. In FIGS. 15-20, however, the grid numbers 1-36 are showninstead of the sector voter margins, for ease of reference.

Players may take alternating turns beginning with the starting player.During a player's turn, he/she (A) makes one move and then (B) recordsthe move on the scoreboard. Forfeiting a turn (i.e. passing on a turn)is not allowed.

All moves must be of type 1, 1A, 2, 2A, 3, or 3A (described below).Moves of type 1 and 1A establish a new district. These moves are incategory E. Moves of type 2 and 2A expand an existing (i.e. alreadyestablished) district. These moves are in category X. Moves of type 3and 3A resize two adjacent districts. These moves are in category R. “A”means “alternate move.”

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than nine districts have been established. In stage 2,only moves of type 1A, 2, and 3 are allowed. Play enters stage 3immediately after the 9^(th) district is established. In stage 3, thenext move must be of type 2 or 3A if a move of type 2 exists. Otherwise,the next move must be of type 2A or 3A. In many games, stage 2 isskipped and play proceeds directly from stage 1 to stage 3. Playconcludes when no legal moves exist.

The six types of legal moves are as follows. FIGS. 15-20 illustratedifferent examples when certain move types are allowed by the rules.Explanations for the asterisked terms are provided at the end of thesedescriptions.

-   -   1 Establish a new district by placing its home base marker 600        on a vacant sector 504. This move must meet two        requirements. (a) The sector must be 2 or more        (horizontal+vertical) steps away from each previously placed        home base marker. The two sectors shown in (i), (ii), and (iii)        in FIG. 18 are 1, 2, and 2 steps away from each other        respectively. Sectors 20 and 29 in FIG. 12 are four steps away        from each other. (b) There must be space to grow this district        to a size of 4 connected* sectors.        -   The position shown in FIG. 15A illustrates moves of type 1.            Here, placing a home base marker on sector 10, 11, or 17 is            not allowed because it violates requirement (a). Also,            placing a home base marker on sector 30, 35, or 36 is not            allowed because it violates requirement (b). In this            position there are only two possible moves of type 1: place            a home base marker on sector 12 or sector 31.    -   1A Establish a new district by placing its home base marker 600        on a vacant sector 504. This move must meet two        requirements. (c) The sector must be in the largest open space        on the board. An open space is a set of connected* vacant        sectors. (d) The sector must be the farthest (in number of        steps) from a previously placed home base marker (among the        sectors satisfying requirement (c)).        -   The position shown in FIG. 15B illustrates moves of type 1A.            Here, placing a home base marker on sector 3 or 17 is not            allowed because it violates requirement (c). Also, placing a            home base marker on sector 30 or 35 is not allowed because            it violates requirement (d). In this position there are only            three possible moves of type 1A: place a home base marker on            sector 14, 19, or 36.    -   2 Expand an existing district by placing one of its expansion        markers 602 on a vacant sector 504. This move must meet three        requirements. (e) The district must remain connected.* (f) The        district's new size after this move—including the new sector and        any sectors that are captured**—may not exceed 4 sectors. (g) No        district may be trapped.***        -   The position shown in FIG. 16A illustrates moves of type 2.            Here, placing an orange expansion marker 602 on sector 3 or            23 is not allowed because (e) is violated. Also, placing a            yellow expansion marker 602 on sector 10 is not allowed            because (f) is violated. Also, placing a purple expansion            marker 602 on sector 7 or 13 is not allowed because            sector(s) are captured and (f) is violated. Finally, placing            a gray expansion marker 602 on sector 30 is not allowed            because the Green District would be trapped and (g) would be            violated.    -   2A Expand an existing district by placing one of its expansion        markers 602 on a vacant sector 504. This move must meet three        requirements. (e) The district must remain connected.* (h) Only        the smallest expandable district may be expanded. A district is        expandable if there is at least one vacant sector adjacent to        it. (i) No sectors may be captured.**        -   The position shown in FIG. 16B illustrates moves of type 2A.            Here, placing a blue expansion marker on sector 6 is not            allowed because (e) is violated. Also, placing an orange            expansion marker on sector 35 is not allowed because (h) is            violated. Finally, placing a blue expansion marker on sector            35 is not allowed because (i) would be violated. In this            position there are only five possible moves of type 2A:            place a brown expansion marker on sector 6; place a gray            expansion marker on sector 6 or 24; or place a blue            expansion marker on sector 24 or 36.    -   3 Reassign a community from one district (say District X) to        another (say District Y) by removing the District X marker from        a sector and replacing it with a District Y expansion marker.        This move must meet five requirements. (j) District Y must exist        prior to this move. (k) District Y must not be expandable prior        to this move. (l) District X must be at least 2 sectors larger        than District Y prior to this move. (m) Districts X and Y must        each remain connected.* (n) The District X marker that is        removed must be an expansion marker; it may not be a home base        marker.        -   The position shown in FIG. 17A illustrates moves of type 3.            Here, reassigning sector 13 to the Red District is not            allowed because the Red District has not been established            and (j) is violated. Reassigning sector 16 to the Gray            District is not allowed because the Gray District is            expandable and (k) is violated. Reassigning sector 9 to the            Green District is not allowed because (l) is violated.            Reassigning sector 23 or 35 to the Pink District is not            allowed because (m) is violated. Reassigning sector 24 or 36            to the Pink District is not allowed because (n) is violated.            In this position there are only two possible moves of type            3: reassign sector 16 to the Green District or reassign            sector 28 to the Pink District.    -   3A This move has the same requirements as move type 3 except        that (n) is not required        -   The position shown in FIG. 17B illustrates moves of type 3A.            In this position, six moves of type 3A are available:            reassign sector 26 or 31 to the Red District; reassign            sector 26 to the Purple District; and reassign sector 16,            23, or 27 to the Pink District.

-   * See subsection entitled “Connectedness” below

-   * See subsection entitled “Captured sectors” below

-   ** See subsection entitled “Trapped districts” below

Connectedness

Two sectors are adjacent—and connected—if and only if they share acommon edge. For example, the two-sector area shown in (i) in FIG. 18 isconnected, but the two-sector areas shown in (ii) and (iii) in FIG. 18are NOT connected.

In this game, every political district must be connected at all times.That is, at all times and for any two sectors that belong to a givendistrict (say District X), there must be a path within District X—asequence of adjacent sectors that all belong to District X—connectingthose two sectors.

Captured Sectors

A set of connected, vacant sectors is captured if it is (i) surroundedby a single district or (ii) surrounded by the edge of the board on oneside and a single district on the other side. In FIG. 19, sector 1 iscaptured by the Green District (consisting of sectors 2 and 7-8);sectors 5-6 are captured by the Blue District; sectors 31-36 arecaptured by the Red District; and sector 16 is captured by the BlueDistrict. No other sectors are captured.

A move of type 2 which captures exactly one sector is allowed if thedistrict's new size—including the sector on which the marker is placedand the sector that is captured—is no greater than four sectors. Allother moves that capture sectors are forbidden. For example, if sector 1is vacant, it is permissible to add sector 2 to a district consisting ofsectors 7-8. In this case, sector 1 is captured and the new districtconsists of sectors 1-2 and 7-8. However, if sectors 5-6 are vacant,adding sector 12 to a district consisting of sectors 4 and 10-11 is notallowed.

A sector that is captured during a legal move of type 2 is immediatelyassigned to the district that has captured it. An expansion marker isimmediately placed on this sector, and the scoreboard is updatedappropriately.

Trapped Districts

A district is trapped if (i) it (and the open spaces beside it) iseither surrounded by a single district or is surrounded by the edge ofthe board on one side and a single district on the other side and (ii)its size (in sectors) plus the sizes of the open spaces beside it isless than four.

In FIG. 20, the Gray District (consisting of sector 1) is trapped by theGreen District (consisting of sectors 2 and 7-8) and the Pink and OrangeDistricts are trapped by the Blue District. The Yellow District is nottrapped because it can still grow to a size of four sectors.

A move of type 2 which traps a district is forbidden. For example, ifthe Gray District consists of sector 1 and the Green District consistsof sectors 7-8, then an expansion of the Green District to sector 2 isnot allowed. Also, placing a blue expansion marker on sector 15 toachieve the position in FIG. 20 is not allowed.

End of Phase 1

Phase 1 ends when no legal moves exist. When this happens, exactly onemarker will occupy each sector, and the state will be partitioned intonine political districts that average four sectors each.

Phase 2: Run an Election Summary

This optional phase of the game accounts for the surprises that canhappen in real-world elections. Sometimes the candidate whose party hasthe majority of voters in a district is defeated by his/her opponent.This may happen if a candidate lacks charisma, public speaking skills,good looks, or other personal qualities or if the candidate takesunpopular stands on issues such as education, health care, the economy,infrastructure, foreign affairs, the environment, etc. In this phase ofthe game, the voter margin in each district is converted into anumerical likelihood of each party winning the district, and an electionin each district is simulated by rolling dice 618.

Details

Each district is considered one at a time beginning with the BrownDistrict.

First, using the table below, the voter margin for the party with morevoters in the district is converted into a numerical likelihood of thatparty winning an election in the district. For example, a “+8 Red” votermargin in the Yellow District converts to a 97% chance for the Red Partyto win an election in the Yellow District.

District Voter Winning Margin Likelihood 0 50% +1 60% +2 69% +3 77% +484% +5 90% +6 93% +7 95% +8 97% +9 99% +10 or more 100% 

Second, a random number from 1-100 is produced by simultaneously rollingthe two 10-sided dice. The result shown on the black (white) die is thevalue of the tens (ones) digit of the random number. For example, if theblack (white) die shows 7 (1), the result is 71. If the black (white)die shows 0 (8), the result is 8. The only exception to the above ruleis that a roll of “zero-zero” gives the result of 100.

Third, the random number is compared to the winning percentage (e.g. 97for the above case). If the random number is less than or equal to thewinning percentage, the party with more voters in the district wins thedistrict election. If the random number is greater than the winningpercentage, the party with fewer voters in the district wins thedistrict election. In the above example, the Red Party wins the YellowDistrict election if the random number is from 1-97, and the Blue Partywins the Yellow District election if the random number is from 98-100.If both parties have a 50% chance of winning a district, the Blue Partywins if the random number is from 1-50 and the Red Party wins if therandom number is from 51-100. If a party has a 100% chance of winning adistrict, it automatically wins that district without a dice roll. Afterthe winner of an election is identified, the scoring token that matchesthe district color is placed on the “Blue Wins” or “Red Wins” square inthat district's portion of the scoreboard 604.

The above procedure is repeated for each of the nine districts.

Phase 3: Identify the Winner

In the game's final phase, the overall winner is identified.

If phase 2 is played, the winner is the player whose party wins five ormore district elections. If phase 2 is not played, players identify theparty that controls each district, i.e. the party with more voters ineach district. This is done by looking at the positions of the scoringtokens on the scoreboard. The winner is the player whose party controlsmore districts than his/her opponent. If the players control an equalnumber of districts, the result is a tie.

Example of Play

FIGS. 21-30 provide an example of play. After the setup is finished,assume that the initial sector arrangement is as shown in FIG. 11.During play, sector grid numbers are not visible to the players. InFIGS. 21-29, however, sector grid numbers 1-36 are shown for ease ofreference.

During stage 1 of play, only moves of type 1, 2, and 3 are allowed.After 15 turns, assume the position in FIG. 21 is reached. An guide tothe specific home base markers 600 and expansion markers 602 shown isprovided in Table 1.

The available moves of type 1 in this position are as follows:

-   -   Establish a new district on sector 1, 13, 15, 18, 21, 22, 24,        27, 34, or 36

The available moves of type 2 in this position are as follows:

-   -   Expand Red District to sector 5    -   Expand Orange District to sector 19 or 31    -   Expand Yellow District to sector 17 or 18    -   Expand Blue District to sector 22, 23, 24, 27, or 36    -   Expand Purple District to sector 1, 3, 7, 9, or 14    -   Expand Pink District to sector 3, 5, 9, 15, 17, or 22    -   Expand Gray District to sector 14, 21, 27, 31, or 33

No moves of type 3 are available in this position.

Notes:

-   -   Sector 1 is captured and immediately added to the Purple        District if the Purple District is expanded to sector 7.    -   An expansion of the Blue District to sector 34 or 35 is not        allowed because it violates requirement (f).    -   An expansion of the Yellow District to sector 5 is not allowed        because the Red District would be trapped and requirement (g)        would be violated.    -   An expansion of the Gray District to sector 19 is not allowed        because the Orange District would be trapped and requirement (g)        would be violated.

Eight moves later in the game, after a total of 23 moves, the newposition is shown in FIG. 22. The game is still in stage 1 because amove of type 1 is available.

The available moves of type 1 in this position are as follows:

-   -   Establish Green District on sector 21, 22, or 27

The available moves of type 2 in this position are as follows:

-   -   Expand Brown District to sector 33 or 35    -   Expand Orange District to sector 19

No moves of type 3 are available in this position.

Notes:

-   -   Reassigning sector 4 from the Pink District to the Red District        is not allowed because sector 4 is occupied by a home base        marker (see requirement (n)).    -   Similarly, reassigning sector 11 from the Yellow to Red District        is not allowed.    -   Reassigning sector 12 from the Yellow District to the Red        District is not allowed because the Yellow District would not be        connected (see requirement (m)).

Four moves later, after a total of 27 moves, the new position is shownin FIG. 23. Here, (i) there is no way to make a move of type 1 thatsatisfies its criteria and (ii) fewer than nine districts have beenestablished. Thus, stage 2 of play may begin. The next move must be oftype 1A, 2, or 3.

The available moves of type 1A in this position are as follows:

-   -   Establish Green District on sector 22

The available moves of type 2 in this position are as follows:

-   -   Expand Orange District to sector 13

No moves of type 3 are available in this position.

Notes:

-   -   A move of type 1A requires that a new district be established on        a sector that is in the largest open space on the board. Among        the sectors satisfying this requirement, a sector that ties for        being the most steps away from a previously placed home base        marker must be selected. In the current position, the largest        open space consists of three sectors: 17, 22, and 23. Among        these sectors, only one—sector 22—is two or more steps away from        all previously placed home base markers. So there is only one        legal move of type 1A in this position.

Two moves later, after a total of 29 moves, the new position is shown inFIG. 24. Here, all nine districts have been established, so we are instage 3 of play. In stage 3, the next move must be of type 2 or 3Awhenever a move of type 2 exists. Otherwise, the next move must be oftype 2A or 3A.

The available moves of type 2 in this position are as follows:

-   -   Expand Orange District to sector 14    -   Expand Green District to sector 23

The available moves of type 3A in this position are as follows:

-   -   Reassign sector 4 from Pink District to Red District    -   Reassign sector 11 from Yellow District to Red District

Notes:

-   -   The Red District is the only district that is not expandable.        Thus, it is the only district that could possibly “steal” a        community from another district (see requirement (k)).

Three moves later, after a total of 32 moves, the new position is shownin FIG. 25. In this position, no move of type 2 exists. Thus, the nextmove must be of type 2A or 3A.

The available moves of type 2A in this position are as follows:

-   -   Expand Brown District to sector 35    -   Expand Blue District to sector 35    -   Expand Purple District to sector 3 or 9    -   Expand Pink District to sector 3 or 9

The available moves of type 3A in this position are as follows:

-   -   Reassign sector 4 from Pink District to Red District    -   Reassign sector 11 from Yellow District to Red District

Notes:

-   -   Four districts—Brown, Blue, Purple, Pink—currently tie for being        the smallest expandable district (see requirement (h)).

Two moves later. After a total of 34 moves, the new position is shown inFIG. 26. In this position, a move of type 2 is available. Thus, the nextmove must be of type 2 or 3A.

The available moves of type 2 in this position are as follows:

-   -   Expand Red District to sector 3

The available moves of type 3A in this position are as follows:

-   -   Reassign sector 9 from Purple District to Pink District

One move later, after a total of 35 moves, the new position is shown inFIG. 27. Here, no move of type 2 exists. Thus, the next move must be oftype 2A or 3A.

The available moves of type 2A in this position are as follows:

-   -   Expand Brown District to sector 35    -   Expand Blue District to sector 35

The available moves of type 3A in this position are as follows:

-   -   Reassign sector 9 from Purple District to Pink District

One move later, after a total of 36 moves, the new position is shown inFIG. 28. There are no vacant sectors, so all future moves will be oftype 3A. Play continues until no such moves exist.

The available moves in this position are as follows:

-   -   Reassign sector 21 from Brown District to Green District    -   Reassign sector 21 from Brown District to Pink District    -   Reassign sector 9 from Purple District to Pink District

One move later, after a total of 37 moves, the new position is shown inFIG. 29. No legal moves exist in this position, so phase 1 of playconcludes. The districts that have been formed are now final.

The final position at the end of phase 1 is shown in FIG. 30, with thedistricts 510-526 marked. The markers played and the community votermargins are shown.

The final district voter margins are shown on the scoreboard (see tablebelow).

If phase 2 is not played, the game immediately ends, and the Blue Partywins by a score of 5 districts to 3 districts (with one tied district).

No. Voter District Communities Margin 1. Brown 4 0 (=6 + 3 − 5 − 4) 2.Red 4 +7 Blue (=0 + 8 + 6 − 7) 3. Orange 4 +3 Blue (=5 + 3 − 3 − 2) 4.Yellow 4 +8 Red (=5 + 9 − 2 − 4) 5. Green 3 +5 Red (=6 + 7 − 8) 6. Blue4 +2 Blue (=9 + 2 − 1 − 8) 7. Purple 5 +9 Red (=1 + 2 + 8 − 1 − 1) 8.Pink 4 +1 Blue (=7 + 4 − 6 − 4) 9. Gray 4 +9 Blue (=7 + 0 + 5 − 3)

If phase 2 is played, dice 618 are rolled to determine the winning partyin each district. In the game at hand, the final voter margin of the(Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) Districtis (0, +7 Blue, +3 Blue, +8 Red, +5 Red, +2 Blue, +9 Red, +1 Blue, +9Blue). Using a preceding table, these margins translate to winninglikelihoods of (50%, 95%, 77%, 97%, 90%, 69%, 99%, 60%, 99%) for theparties with the majority of voters in these districts respectively.Note that each party has a 50% chance of winning the Brown District, andno party automatically wins a district with 100% probability.

Dice 618 are then thrown to determine the election results. The resultsare summarized in the table below. Despite being at a disadvantage goinginto the election, the Red Party “gets lucky” and wins the elections infive out of nine districts. The Red Party wins the game by a score of 5districts to 4 districts.

Voter Winning Dice Election District Margin Likelihood Roll Result Brown0 50% for Blue 68 Red Wins Red +7 Blue 95% for Blue 95 Blue Wins Orange+3 Blue 77% for Blue 4 Blue Wins Yellow +8 Red  97% for Red  25 Red WinsGreen +5 Red  90% for Red  41 Red Wins Blue +2 Blue 69% for Blue 13 BlueWins Purple +9 Red  99% for Red  92 Red Wins Pink +1 Blue 60% for Blue61 Red Wins Gray +9 Blue 99% for Blue 30 Blue Wins

Rules for a Symmetric Game

A starting region with a large connected portion of high-numberedsectors favoring the Blue Party but no large connected portion ofhigh-numbered sectors favoring the Red Party is biased in favor of theRed Party. In such a setting, the player representing the Red Party willmore easily be able to concentrate or “pack” the voting power of theopposing party into a small number of districts than the playerrepresenting the Blue Party. Thus, the Red Party is more likely to winthe game.

The purpose of a symmetric game is to remove bias from the initialsector arrangement and give each party—Red and Blue—a fair chance ofwinning the game. This is particularly important in a tournamentsetting.

A symmetric game has three additional rules compared to a regular game.Rule 1 creates a symmetric initial sector arrangement, and rules 2 and 3minimize the possibility of a symmetric position during play. The threerules are as follows.

-   -   1. During the game setup, the sector arrangement must be        counter-symmetric with respect to an imaginary dot in the center        of the region (see FIG. 12). In a counter-symmetric sector        arrangement the two sectors comprising every pair of        diametrically-opposed sectors—every two sectors that are on        exact opposite sides of the state—have the same voter margins        but they favor different parties. This arrangement guarantees        that the starting map is unbiased, favoring neither party.        -   FIG. 31 shows a counter-symmetric sector arrangement. Three            portions of the region 502 are shown: the central square            622; the inner ring 624 which encircles it; and the outer            ring 626 which encircles the inner ring. Bold lines            distinguish these three portions of the region. Note that            the sectors on opposite sides of the central square have the            same number but opposite voter margins. Sector 15 is “+6            Red” whereas sector 22 is “+6 Blue,” and sector 16 is “+7            Blue” whereas sector 21 is “+7 Red” (see FIG. 12 for the            sector number key). The sectors on opposite sides of the            inner ring 624 also have the same number but opposite voter            margins. The same holds true for the sectors in the outer            ring 626. This map's symmetry gives each player a fair            chance of winning the game.        -   A random counter-symmetric initial sector arrangement can be            efficiently created using the “Symmetric Game Setup Table”            628 shown in FIG. 32 and any thirty six expansion markers.            The procedure works as follows. First, all markers are            removed from the Symmetric Game Setup Table. The thirty-six            sector tiles are then mixed and organized face down into a            single deck. Sector tiles are drawn from the deck one at a            time. When a sector tile is drawn, players look at the            portion of the Symmetric Game Setup Table 628 that matches            the sector's voter margin and color (e.g. “+5” and “Blue”).            If the total number of markers in this portion of the table            is greater than or equal to the total number of markers in            the portion of the table with the same voter margin but            opposite color, the sector tile is placed face up in the            first unoccupied location according to the sector sequence            in FIG. 12. Otherwise the sector tile is placed face up in            an unoccupied location that is diametrically opposed to            where an opposing sector tile (with the opposite voter            margin) has already been placed. Then a marker is placed on            a dot in the Symmetric Game Setup Table that matches the            sector's voter margin and color. This continues until all 36            sector tiles are drawn and placed face up.    -   2. During the second and third moves of phase 1 (the 1^(st) move        made by the player who goes second, and the 2^(nd) move made by        the player who goes first), no marker may be placed on a sector        that is diametrically opposed to a sector on which a marker has        already been placed.    -   3. During phase 1, a move (of any type) that creates a district        that (a) has size four and (b) coincides with the central square        is never allowed.

Tournament Play

This example of the present technology a game of pure skill if (A)players decide who plays first prior to the start of the game, (B) asymmetric game is played, and (C) phase 2 of the game is skipped. Thisform of the game, like international chess and the Japanese game go, ishighly suited to tournament play. Unlike chess and go, the initial boardposition in this game is always different, so every game has a uniqueopening.

Handicap Play

This game is suited to handicap play. If the players' skill levelsdiffer, the playing field can be leveled by changing the voter margin ofone or more sectors. For example, if the stronger player represents theBlue Party, the players may agree, before any sector tiles 610 areplaced, to change the voter margin of the first “+8 Blue” sector tilethat is placed from “+8 Blue” to 0. Alternatively, the weaker player maybe allowed to make more than 50% of the moves—for example 5 of every 9moves.

Game Alternative #1: Form Seven Districts (Each of Size 5)

In one variation of strategy game 500, seven districts are formedinstead of nine districts. In this variation, only seven marker subsets620, representing seven district colors, are used, and the average sizeof a district at the end of the game is about five sectors 504. At theend of the game, the region will be divided into seven politicaldistricts. This variation of the game may be played according to thesame rules above except that the requirements for moves of type 1, 1A,and 2 are slightly different as described below:

-   -   A move of type 1 must meet two requirements. (a) The sector on        which the home base marker is placed may not be in the ring        surrounding any previously placed home base marker. (b) There        must be space to grow the new district to a size of 5 connected        sectors. In FIG. 12, the ring surrounding sector 1 consists of        sectors 2 and 7-8. The ring surrounding sector 10 consists of        sectors 3-5, 9, 11, and 15-17. The ring surrounding sector 5        consists of sectors 4, 6, and 10-12.    -   A move of type 1A must meet two requirements. (c) The sector on        which the home base marker is placed must be in the largest open        space on the board. (d) Among the sectors satisfying the        requirement c, the sector must be one that is not in a ring        surrounding a previously placed home base marker. If no such        sector exists, the sector must satisfy requirement c and be the        farthest (in number of horizontal+vertical steps) from a        previously placed home base marker.    -   A move of type 2 must meet three requirements. (e) The district        must remain connected. (f) The district's new size—including the        new sector and any captured sectors—must not be greater than 5        sectors. (g) No district may be trapped. A district is trapped        if (i) it (and the open spaces beside it) is either surrounded        by a single district or is surrounded by the edge of the board        on one side and a single district on the other side and (ii) its        size (in sectors) plus the sizes of the open spaces beside it is        less than 5.

Example 4

Strategy games of Example 4 have a taxonomic code A/G/S2/P/2/12/U/2/EXR.They are analog, multi-player games with 72 square sectors and apolitical focus in which two types of elements—namely two politicalparties—are present and twelve districts are formed. The games proceedaccording to alternating, turn-based play; there are two players; andmoves in categories “E,” “X,” and “R” are allowed.

One example is strategy game 700 as shown in FIGS. 33-35, which is alarger version of the game described in Example 3. Strategy game 700includes a region 702, which as illustrated is set up as an 8×9 griddivided into sector placeholders 718 numbered 1-72. Sector tiles 716 areprovided, which may be shuffled or otherwise reorganized, and laid outon the region with one sector tile 716 per sector placeholder 718 toform 72 sectors 704 in the region 702 at the start of a game, as shownin FIG. 34. The set of elements 724 for each sector 704 is provided oneach sector tile 716.

Synopsis

Strategy game 700 is very similar to strategy game 500 described inExample 3. The main differences are as follows. First, in strategy game700, there are 72 sectors—exactly twice as many sectors of each kind asin strategy game 500. Second, at the start of strategy game 700 thesector tiles 716 are randomly placed in a 9×8 rectangular arrangementwithin region 702 to form sectors 704. Third, twelve districts—theaverage size of which at the end of the game may be six sectors 704—willbe formed. A set of markers 726 containing a total of 12 marker subsets710 may be used, each marker subset 710 representing a district andconsisting of one home base marker 712 and eighteen expansion markers714. As shown there are nine rectangular home base markers (one of eachfor Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray) and162 square expansion markers (18 of each of the same colors as therectangular home base markers), and then three additional home basemarkers that are diamond shaped (one each for the Brown2, Red2, andOrange2 Districts) and three additional sets of expansion markers thatare triangular in shape (18 of each of the same colors as the diamondshaped home base markers). Each district is distinguishable by the colorand/or shape of the markers used to form it. Fourth, the precise rulesfor making moves of types 1, 1A, 2, and 2A are slightly different inthis game to encourage most districts to have a size of six sectors 704at the end of the strategy game 700.

In the illustrated example, strategy game 700 includes two scoreboards,a first scoreboard 706 and a second scoreboard 708. Each scoreboard hasa plurality of rows 722 and looks like FIG. 13. The total number of rowsin both scoreboards is at least 12, and thus one row may be used totrack the voter margin of each district. There are also twelve scoringtokens 720, one for each district. In other examples of strategy game700, there may only be one scoreboard, which would have one row 722 foreach of the twelve districts to be formed during the game.

As illustrated, FIG. 34 shows the region 702 at the start of strategygame 700, and FIG. 35 shown one possible solution at the end of gameplay. In FIG. 35, the home base markers 712 and expansion markers 714have been placed in a manner that establishes twelve districts 728-750.

The table below shows the final result of this game. The Blue Party winsthis example game by a score of 6 districts to 5 districts (with onetied district).

Voter District Margin Brown +2 Blue Red +13 Blue  Orange +5 Red  Yellow+4 Red  Green +9 Red  Blue +2 Blue Purple +5 Blue Pink +4 Blue Gray +2Red  Brown2 0 Red2 +4 Blue Orange2 +10 Red  

Playing the Game

Strategy game 700 may have the same three phases as, and may be playedin a manner that is nearly identical to, Example 3.

Phase 1: Build Political Districts

During a player's turn, he/she (A) makes one move and (B) records themove on the appropriate scoreboard 706 or 708. All moves must be of type1, 1A, 2, 2A, 3, or 3A below. About 80 moves—40 by each player—are madein a game. The game ends when no legal moves exist. The winner is theplayer whose party controls more districts than his/her opponent.

Play may be divided into three stages. In stage 1, only moves of type 1,2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1exist and (ii) fewer than 12 districts have been established. In stage2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3immediately after the 12^(th) district is established. In stage 3, thenext move must be of type 2 or 3A if a move of type 2 exists. Otherwise,the next move must be of type 2A or 3A. In many games, stage 2 isskipped and play proceeds directly from stage 1 to stage 3. Playconcludes when no legal moves exist. Forfeiting a turn (i.e. passing ona turn) is not allowed.

The rules may provide six types of legal moves, such as those listedbelow. The terms “captured” and “connected” have the same meaning asdescribed in Example 3. Moves of type 3 and 3A are identical to Example3.

-   -   1 Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (a) The        sector may not be in the ring surrounding any previously placed        home base marker. In FIG. 33, the ring surrounding sector 1        consists of sectors 2 and 9-10. The ring surrounding sector 10        consists of sectors 1-3, 9, 11, and 17-19. The ring surrounding        sector 5 consists of sectors 4, 6, and 12-14. (b) There must be        space to grow this district to a size of 6 connected sectors.    -   1A Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (c) The        sector must be in the largest open space on the board. An open        space is a set of connected vacant sectors. (d) Among the        sectors satisfying requirement c, the sector must be one that is        not in a ring surrounding a previously placed home base marker.        If no such sector exists, the sector must satisfy requirement c        and be the farthest (in number of horizontal+vertical steps)        from a previously placed home base marker. Sectors 2 and 21 in        FIG. 33 are five steps away from each other.    -   2 Expand an established district by placing one of its expansion        markers on a vacant sector. This move must meet three        requirements. (e) The district must remain connected. (f) The        district's new size—including this new sector and any captured        sectors—must not be greater than 6 sectors. (g) No district may        be trapped. A district is trapped if (i) it (and the open spaces        beside it) is either completely surrounded by a single district        or is surrounded by the edge of the board on one side and a        single district on the other side and (ii) its size (in sectors)        plus the sizes of the open spaces beside it is less than 6.    -   2A Expand an established district by placing one of its        expansion markers on a vacant sector. This move must meet four        requirements. (e) The district must remain connected. (h) Only        the smallest expandable district may be expanded. A district is        expandable if there is at least one vacant sector adjacent to        it. (i) No sectors may be captured. (o) The district's new size        must not be greater than 19 sectors. This last requirement        relates to limited marker quantities.    -   3 Reassign a community from one district (say District X) to        another (say District Y) by removing the District X marker from        a sector and replacing it with a District Y expansion marker.        This move must meet five requirements. (j) District Y must exist        prior to this move. (k) District Y must not be expandable prior        to this move. (l) District X must be at least 2 sectors larger        than District Y prior to this move. (m) Districts X and Y must        each remain connected after this move. (n) The District X marker        that is removed must be an expansion marker; it may not be a        home base marker.    -   3A This move has the same requirements as move type 3 except        that (n) is not required.

End of Phase 1

Phase 1 ends when no legal moves exist. When this happens, exactly onemarker will occupy each sector, and the state will be partitioned into12 political districts that average 6 sectors each. FIG. 35 shows apossible position at the end of phase 1.

Phase 2: Run an Election

This phase of the game is nearly identical to Example 3 except that adifferent table (shown below) is used to convert a district's votermargin into the probability that the party with more voters in thedistrict wins an election in the district.

District Voter Winning Margin Likelihood 0 50% +1 59% +2 67% +3 74% +480% +5 85% +6 89% +7 92% +8 94% +9 96% +10 98% +11 99% +12 or more 100% 

Phase 3: Identify the Winner

This phase of the game is exactly the same as in Example 3. If phase 2is played, the winner is the player whose party wins seven or moredistrict elections. If each party wins six district elections, theresult is a tie. If phase 2 is not played, players look at the scoringtokens on the scoreboard to identify the party that controls eachdistrict, i.e. the party with more voters in each district. The winneris the player whose party controls more districts than his/her opponent.If the players control an equal number of districts, the result is atie.

Game Alternative #1: Form 18 Districts (with an Average Size of 4Sectors)

One variation of strategy game 700 includes the formation of 18districts—instead of 12—in the same 9×8 region 702. In this variation,exactly twice as many home base, expansion, and scoring tokens are usedcompared to Example 3. The size of the average district at the end ofthis variation of the game is 4 sectors 704. This variation may beplayed according to the same rules above except that the requirementsfor moves of type 1, 1A, and 2 may be slightly different as describedbelow:

-   -   A move of type 1 must meet two requirements. (a) The sector on        which a home base marker is placed must be at least 2        horizontal+vertical steps away from all previously placed home        base markers. (b) There must be space to grow the new district        to a size of 4 connected sectors. In FIG. 33, sectors 20 and 54        are six steps away from each other.    -   A move of type 1A must meet two requirements. (c) The sector on        which a home base marker is placed must be in the largest open        space on the board. (d) Among the sectors satisfying the        requirement c, the sector must tie for being the farthest (in        number of horizontal+vertical steps) from a home base marker.    -   A move of type 2 must meet three requirements. (e) The district        must remain connected. (f) The district's new size—including the        new sector and any captured sectors—must not be greater than 4        sectors. (g) No district may be trapped. A district is trapped        if (i) it (and the open spaces beside it) is surrounded by the        edge of the board on one side and a single district on the other        side and (ii) its size (in sectors) plus the sizes of the open        spaces beside it is less than 4.

Rules for a Symmetric Game

As with strategy game 500, strategy game 700 can be played as asymmetric game. The purpose of a symmetric game is to remove bias fromthe initial sector arrangement and give each party—Red and Blue—a fairchance of winning the game.

When strategy game 700 is played as a symmetric game, there are twoadditional rules that may be used compared to a regular game. Rule 1creates a symmetric initial sector arrangement, and rule 2 reduces thepossibility of a symmetric position during play. The two rules are asfollows.

-   -   1. During the game setup, the sector arrangement must be        counter-symmetric with respect to an imaginary dot in the center        of the state. This arrangement guarantees that the starting map        is unbiased, favoring neither party. A procedure for doing this        may be similar or nearly identical to that described in the        subsection “Rules for a symmetric game” in the description of        Example 3.    -   2. During the 2^(nd), 3^(rd), 4^(th), and 5^(th) moves of phase        1 (i.e. the 1^(st) and 2^(nd) moves made by the player who goes        second, and the 2^(nd) and 3^(rd) moves made by the player who        goes first), no marker may be placed on a sector that is        diametrically opposed to a sector on which another marker has        already been placed.

Example 5

Strategy games of Example 5 have a taxonomic code A/G/S3/P/2/15/U/2/EXR.They are analog, multi-player games with 90 square sectors and apolitical focus in which two types of elements—namely two politicalparties—are present and 15 districts are formed. These games proceedaccording to alternating, turn-based play; there are two players; andmoves in categories “E,” “X,” and “R” are allowed.

Strategy games of Example 5 are is very similar to the games describedin Examples 3-4. In one example, two players—Red and Blue—vie forpolitical control of a 9×10 rectangular state by competitively creating15 political districts (whose average size is 6) out of 90 squarecommunities. The game can be played with any 90 sectors in which thesets of red and blue sectors are identical—for example 5 each of sectortiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (80 sector tiles);3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 4sector tiles with a voter margin of 0. Players take alternating turnsbeginning with the starting player. During a player's turn, he/she (A)makes one move and then (B) records the move on two scoreboards. Allmoves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 100 moves—50by each player—are made in a game. The game ends when no legal movesexist. The winner is the player whose party controls more districts thanhis/her opponent.

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than 15 districts have been established. In stage 2, onlymoves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediatelyafter the 15^(th) district is established. In stage 3, the next movemust be of type 2 or 3A if a move of type 2 exists. Otherwise, the nextmove must be of type 2A or 3A. Play concludes when no legal moves exist.

The six types of moves allowed in the rules for games of this type aresummarized below.

-   -   1 Establish a district on a sector that is not in the ring        surrounding any other home base marker. There must be space to        grow this district to size 6 or more.    -   1A Establish a district on a sector that is not in the ring        surrounding any other home base marker (among the sectors in the        largest open space on the board). If this is not possible,        establish a district on a sector that is farthest (in number of        steps) from a home base marker (among the sectors in the largest        open space on the board).    -   2 Expand a district so (e) it remains connected, (f) its new        size (including any captured sectors) is 6 or less, and (g) no        district is trapped.    -   2A Expand the smallest expandable district so (e) it remains        connected, (i) no sectors are captured, and (o) its new size is        19 or less.    -   3 (Same as in Example 3).    -   3A (Same as in Example 3).

Example 6

Strategy games of Example 6 are larger versions of the games describedin Examples 3-5. These games have a taxonomic codeA/G/S4/P/2/15/U/2/EXR. They are analog, multi-player games with 121square sectors and a political focus in which two types ofelements—namely two political parties—are present and 15 districts areformed. These games proceed according to alternating, turn-based play;there are two players; and moves in categories “E,” “X,” and “R” areallowed.

In at least one example, two players—Red and Blue—vie for politicalcontrol of an 11×11 square region by competitively creating 15 politicaldistricts (whose average size is just above 8) out of 121 squaresectors, each of which represents a community. The game can be playedwith any 121 sectors in which the sets of red and blue sectors areidentical—for example 7 each of sector tiles “+1 Red” to “+8 Red” and“+1 Blue” to “+8 Blue” (112 sector tiles); 3 each of sector tiles “+9Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with a votermargin of 0. Players take alternating turns beginning with the startingplayer. During a player's turn, he/she (A) makes one move and then (B)records the move on two scoreboards. All moves must be of type 1, 1A, 2,2A, 3, or 3A below. About 140 moves—70 by each player—are made in agame. The game ends when no legal moves exist. The winner is the playerwhose party controls more districts than his/her opponent.

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than 15 districts have been established. In stage 2, onlymoves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediatelyafter the 15^(th) district is established. In stage 3, the next movemust be of type 2 or 3A if a move of type 2 exists. Otherwise, the nextmove must be of type 2A or 3A. Play concludes when no legal moves exist.

The six types of legal moves are summarized below.

-   -   1 Establish a district on a sector that is at least 3        (horizontal+vertical) steps away from all other home base        markers. There must be space to grow this district to size 8 or        more.    -   1A Establish a district on a sector that ties for being the        farthest (in number of horizontal+vertical steps) from a home        base marker (among the sectors in the largest open space on the        board).    -   2 Expand a district so (e) it remains connected, (f) its new        size (including any captured sectors) is 8 or less, and (g) no        district is trapped.    -   2A Expand the smallest expandable district so (e) it remains        connected, (i) no sectors are captured, and (o) its new size is        19 or less.    -   3 (Same as in Examples 3-5)    -   3A (Same as in Examples 3-5)

Example 7

Strategy games of Example 7 are larger versions of the games describedin Examples 3-6. They have a taxonomic code A/G/S5/P/2/21/U/2/EXR. Theyare analog, multi-player games with 169 square sectors and a politicalfocus in which two types of elements—namely two political parties—arepresent and 21 districts are formed. Games of this type proceedaccording to alternating, turn-based play; there are two players; andmoves in categories “E,” “X,” and “R” are allowed.

In one example, two players—Red and Blue—vie for political control of a13×13 region by competitively creating 21 political districts (whoseaverage size is just above 8) out of 169 square sectors. The game can beplayed with any 169 sectors in which the sets of red and blue sectorsare identical—for example 10 each of sector tiles “+1 Red” to “+8 Red”and “+1 Blue” to “+8 Blue” (160 sector tiles); 3 each of sector tiles“+9 Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with avoter margin of 0. Players take turns beginning with the startingplayer. During a player's turn, he/she (A) makes one move and then (B)records the move on three scoreboards. All moves must be of type 1, 1A,2, 2A, 3, or 3A below. About 180 moves—90 by each player—are made in agame. The game ends when no legal moves exist. The winner is the playerwhose party controls more districts than his/her opponent.

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than 21 districts have been established. In stage 2, onlymoves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediatelyafter the 21^(st) district is established. In stage 3, the next movemust be of type 2 or 3A if a move of type 2 exists. Otherwise, the nextmove must be of type 2A or 3A. Play concludes when no legal moves exist.

The six types of legal moves are summarized below.

-   -   1 Establish a district on a sector that is at least 3        (horizontal+vertical) steps away from all other home base        markers. There must be space to grow this district to size 8 or        more.    -   1A Establish a district on a sector that ties for being the        farthest (in number of horizontal+vertical steps) from a home        base marker (among the sectors in the largest open space on the        board).    -   2 Expand a district so (e) it remains connected, (f) its new        size (including any captured sectors) is 8 or less, and (g) no        district is trapped.    -   2A Expand the smallest expandable district so (e) it remains        connected, (i) no sectors are captured, and (o) its new size is        19 or less    -   3 (Same as in Examples 3-6)    -   3A (Same as in Examples 3-6)

Example 8

This is a larger version of the games described in Examples 3-7. Thisgame has taxonomic code A/G/S6/P/2/21/U/2/EXR. It is an analog,multi-player game with 210 square sectors and a political focus in whichtwo types of elements—namely two political parties—are present and 21districts are formed. The game proceeds according to alternating,turn-based play; there are two players; and moves in categories “E,”“X,” and “R” are allowed.

This game is very similar to Examples 3-7. In this game, two players—Redand Blue—vie for political control of a 15×14 rectangular state bycompetitively creating 21 political districts (whose average size is 10)out of 210 square communities. The game can be played with any 210sectors in which the sets of red and blue sectors are identical—forexample 12 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to“+8 Blue” (192 sector tiles); 6 each of sector tiles “+9 Red” and “+9Blue” (12 sector tiles); and 6 sector tiles with a voter margin of 0.Players take turns beginning with the starting player. During a player'sturn, he/she (A) makes one move and then (B) records the move on threescoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below.About 240 moves—120 by each player—are made in a game. The game endswhen no legal moves exist. The winner is the player whose party controlsmore districts than his/her opponent.

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than 21 districts have been established. In stage 2, onlymoves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediatelyafter the 21^(st) district is established. In stage 3, the next movemust be of type 2 or 3A if a move of type 2 exists. Otherwise, the nextmove must be of type 2A or 3A. Play concludes when no legal moves exist.

The six types of legal moves are summarized below.

-   -   1 Establish a district on a sector that is at least 3        (horizontal+vertical) steps away from all other home base        markers. There must be space to grow this district to size 10 or        more.    -   1A Establish a district on a sector that ties for being the        farthest (in number of horizontal+vertical steps) from a home        base marker (among the sectors in the largest open space on the        board).    -   2 Expand a district so (e) it remains connected, (f) its new        size (including any captured sectors) is 10 or less, and (g) no        district is trapped.    -   2A Expand the smallest expandable district so (e) it remains        connected, (i) no sectors are captured, and (o) its new size is        37 or less.    -   3 (Same as in Examples 3-7)    -   3A (Same as in Examples 3-7)

Example 9

Strategy games of Example 9 have a taxonomic code A/G/S1/P/2/9/I. Theyare analog, multi-player games with 36 square sectors and a politicalfocus in which two types of elements—namely two political parties—arepresent and 9 districts are formed. The game paradigm is simultaneousindependent play, in which each player makes one move per turn insequential turns, independently of the other players. Any number ofplayers—two or more—may play.

Game Summary

In one example, illustrated in FIGS. 36-39B as strategy game 800,players compete, optionally under time constraints, to see who can bestcreate the nine political districts of a square region 802, whichrepresents a state. The region 802 consists of 36 square sectorplaceholders 806 that are formed into 36 sectors 804 at the start ofplay, each sector 804 representing a community. The region 802 has anAmerican-style, two-party political system in which one person iselected to represent each political district. At the outset, thedistricts are formless and the players know the political status of eachcommunity (i.e. which party its citizens favor and by how much). Duringthe first phase of the game, players simultaneously and independentlywork on identical copies of the region 802 to create political districtsthat achieve the pre-defined goal of maximizing the political advantageof the Red Party. During the (optional) second phase of the game,players simultaneously and independently work on identical copies of thesame map to create political districts that achieve the pre-defined goalof maximizing the political advantage of the Blue Party. The winner isthe player who does the best job of achieving the pre-defined goalsduring the game.

Components

FIGS. 36-38 show the game components that may be used to play strategygame 800. A timer 814, which may be digital or analog, may be used inexamples of strategy game 800 that are played under time constraints. Inaddition, each player should have a copy of the same game set whichcontains the following items:

-   -   A region 802    -   36 sector tiles 820, which may have the same element sets as the        sector tiles in Example 3    -   A set of expansion markers 822 divided into nine marker subsets        consisting of a plurality of expansion markers 808. In this        instance, 76 expansion markers in the following amounts and        colors: 8 Brown, 10 Red, 8 Orange, 8 Yellow, 8 Green, 10 Blue, 8        Purple, 8 Pink, 8 Gray    -   9 scoring tokens 812 (one for each of the nine districts to be        formed)    -   A scoreboard 810, which may be identical to scoreboard 604    -   A game board 818 (FIG. 37), which may have a duplicate region        824 divided into 36 duplicate sector placeholders 826. The game        board 818 may also have a plurality of rows 828 configured to        allow a player to track aspects of districts formed on the game        board 818.        Setup (about 10 Minutes)

The players decide (yes or no) if phase 2 of the game will be played,and they agree upon a time limit for each phase of the game. The “yes”option with a 10-minute time limit is recommended. (Such a game lastsabout 50 minutes.)

One player may be selected as the leader. All players except the leadermay organize their 36 sector tiles 820 into 19 face-up piles—one pilefor each number+color combination—so that specific sector numbers andcolors can be quickly located. The leader may spread their sector tiles820 out face down, mix them, and organize them face down into a singledeck. The leader may then draw the 36 sector tiles 820 from the deck oneat a time and place them face up with one on each sector placeholder 806of the region 802 to form sectors 804. Each time a sector is drawn, theleader may announce its (a) position in the sequence, (b) color, and (c)number—for example “Sector 1: Red 4,” “Sector 2: Blue 2,” “Sector 3:Zero,” etc.—so that every other player may find the same sector tile 820from his/her game set and place it in the same location in his/herregion 802. When this process ends, each player has a copy of theleader's sector arrangement in his/her region 802. One possiblearrangement of sectors 804 in the region 802 is identical to that shownin FIG. 11.

Scoreboard and Scoring Tokens

Each player may use his/her scoreboard 810 and scoring tokens 812 tokeep track of the voter margins of the districts that he/she createsduring play.

At the end of each phase of the game, the leader or other players mayvisit each player's playing area to ensure that the positions andorientations of his/her scoring tokens properly show the voter marginsof the districts that he/she has created. Each district's scoring token812 should be placed so that (1) the value in the square it occupies onthe scoreboard plus (2) the number on the side of the scoring token thatfaces up equals the district's voter margin. When the scoring tokens 812are cubes—having sides that are unmarked and marked 10, 20, 30, 40, 50respectively—the (unmarked, 10, 20, 30, 40, 50) side of a scoring tokenshould face up if the voter margin of its district is in the range(0-12, 13-22, 23-32, 33-42, 43-52, 53-62) respectively. The scoringtoken 812 is placed on the scoreboard to reflect the voter margin ineach district. For example, if Player 1's Green District containssectors with voter margins “+6 Blue,” “+1 Red,” “+9 Blue,” and “+7Blue,” then the Green District's voter margin is “+21 Blue” (=6+9+7−1)and Player 1's green scoring token should be placed on the square “GreenDistrict Voter Margin=+11 Blue” with its “10” side facing up. If Player2's Gray District contains sectors with voter margins “+3 Blue,” “+1Red,” “+9 Red,” and “+5 Blue,” then the Gray District's voter margin is“+2 Red” (=1+9−3−5) and Player 2's gray scoring token should be placedon the square “Gray District Voter Margin=+2 Red” with its unmarked sidefacing up.

Playing the Game

Play may consist of the following two phases. The second phase isoptional.

-   -   1. Build political districts that maximize the Red Party's        advantage    -   2. Build political districts that maximize the Blue Party's        advantage        Phase 1: Build Political Districts that Maximize the Red Party's        Advantage

The timer 814 is set to the time limit agreed upon by the players, iftime limits are being used. Play begins when the timer starts or theplayers agree to begin.

During the first phase of the game, each player independently useshis/her colored markers to form 9 political districts—Brown, Red,Orange, Yellow, Green, Blue, Purple, Pink, and Gray—on his/her 36sectors 804. Each district is formed by placing four markers, one perturn, that match the district color on four adjacent sectors. Eachplayer's main goal in this phase is to create 9 political districts—i.e.a district plan—in which the Red Party controls as many districts aspossible. A party controls a district if it has the majority—strictlymore than half—of the voters in a district. Each player's secondary goalis to make the “voter margin in the district that the Red Party controlsby the least amount” as high as possible.

The rules may require that each player's district plan must satisfy thefollowing two requirements:

-   -   A. Each sector must belong to exactly one political district.        That is, there must be exactly one expansion marker 808 on each        sector 804.    -   B. Each political district must consist of four connected        sectors 804.

Each player is free to use his/her scoreboard, game board, and markersas desired. It is recommended that each player (a) use the expansionmarkers 808 beside his/her 36 sectors 804 and scoreboard to create andevaluate potential district plans and (b) use the duplicate region 824on his/her game board 818 to store the best district plan that he/shehas found. At the end of this phase, each player's final district planmust be displayed by a set of 36 markers (four per color) that areplaced either on his/her region 802 or on the duplicate region 824 onhis/her game board 818.

When play concludes, the final district plan made by each player isscored. The scoring of each player's final district plan is done by (a)computing the district voter margins, (b) placing scoring tokensappropriately on the scoreboard, and (c) computing the following values:

-   -   1. The number of districts controlled by the Red Party.    -   2. The lowest voter margin in the districts controlled by the        Red Party (“Lowest Voter Margin in Red Districts”).

If a player's district plan violates requirement A or B above, he/shereceives scores of 0 and 1 for items 1 and 2 respectively.

Illustrative Example

One example of the result of phase 1 of a two-player version of game 800is shown in FIGS. 38A and 38B. Bold lines around the districts createdby the expansion markers 808 indicate the final district plans in phase1 for each player.

At this point, Player 1's scoreboard should show that the voter marginof his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray)District is (+3 Red, +2 Red, +10 Red, +18 Blue, +16 Blue, +12 Red, +5Blue, +4 Red, +8 Red). The Red Party controls 6 districts, and thelowest voter margin in those six districts is “+2 Red.” Player 2'sscoreboard should show that the voter margin of his/her (Brown, Red,Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+2 Red, +7Blue, +7 Red, +2 Red, +3 Red, +3 Red, +5 Red, +19 Blue, +4 Red). The RedParty controls 7 districts, and the lowest voter margin in those sevendistricts is “+2 Red.” Overall, Player 2 has done better in this phaseof the game because his/her district plan gives the Red Party control ofmore districts than Player 1's district plan. Each player uses two redmarkers to mark his/her scores for phase 1 in the first two rows 828 ofhis/her game board 818 as shown below.

Game Board Item Player 1 Player 2 1. No. Districts Controlled by RedParty 6 7 2. Lowest Voter Margin in Red Districts 2 2Phase 2: Build Political Districts that Maximize the Blue Party'sAdvantage

At the end of phase 1, players remove all markers from their scoreboardsand game boards, except the two red markers used to mark their finalscores for phase 1 on their game boards. The timer, if used, is then setto the time limit agreed upon by the players. Play of phase 2 is thenstarted.

During phase 2, play proceeds exactly as in phase 1 except that now eachplayer's (i) main goal is to create a district plan in which the BlueParty controls as many districts as possible and (ii) secondary goal isto make the “voter margin in the district that the Blue Party controlsby the least” as high as possible. The rules for making districts arejust as in phase 1.

When the timer goes off, all players cease their activities, disengagefrom their playing areas, and assemble as a group in the middle of theroom. Working as a team, the players together (a) compute the districtvoter margins, (b) place scoring tokens appropriately on the scoreboard,and (c) compute the following for each player's final district plan:

-   -   3. The number of districts controlled by the Blue Party.    -   4. The lowest voter margin in the districts controlled by the        Blue Party (“Lowest Voter Margin in Blue Districts”).

If a player's district plan violates requirement A or B above, he/shereceives scores of 0 and 1 for items 3 and 4 respectively.

The above quantities are remembered by placing two blue markers on theappropriate squares in rows 3 and 4 of each player's game board.

Illustrative Example

One example of the result of phase 2 of the two-player version of game800 is shown in FIGS. 39A and 39B. Bold lines around the districtscreated by the expansion markers 808 indicate the final district plansin phase 2 for each player.

At this point, Player 1's scoreboard should show that the voter marginof his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray)District is (+5 Blue, +1 Blue, +10 Blue, +2 Blue, +1 Blue, +6 Red, +5Blue, +4 Blue, +22 Red). The Blue Party controls 7 districts, and thelowest voter margin in those seven districts is “+1 Blue.” Player 2'sscoreboard should show that the voter margin of his/her (Brown, Red,Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (“+10 Red,”“+15 Blue,” “+6 Blue,” “+3 Blue,” “+25 Red,” “+3 Blue,” 0, “+3 Blue,”“+5 Blue”). The Blue Party controls 6 districts, and the lowest votermargin in those six districts is “+3 Blue.” Overall, Player 1 has donebetter in this phase of the game because his/her district plan gives theBlue Party control of more districts than Player 2's district plan. Eachplayer uses two blue markers to mark his/her scores for phase 2 in rows3-4 of his/her game board 818 as shown below.

Game Board Item Player 1 Player 2 3. No. Districts Controlled by BlueParty 7 6 4. Lowest Voter Margin in Blue Districts 1 3

Identifying the Winner(s)

The winner of the game is identified, by process of elimination, bylooking at the markers in the rows 828 of each player's game board 818.These markers may show the scores for the following items:

-   -   1. Number of districts controlled by the Red Party in phase 1    -   2. Lowest voter margin in the districts controlled by the Red        Party in phase 1    -   3. Number of districts controlled by the Blue Party in phase 2        (if phase 2 played)    -   4. Lowest voter margin in the districts controlled by the Blue        Party in phase 2 (if phase 2 played)

If phase 2 is not played, the winner is identified as follows. First,every player whose district plan does not tie for having the highestscore for item 1 above is eliminated. Next, among the remaining players,every player whose district plan does not tie for having the highestscore for item 2 above is eliminated. Any player who is not eliminatedwins the game.

If phase 2 is played, the winner is identified as follows. First, everyplayer whose district plan does not tie for having the highest sum ofscores for items 1+3 is eliminated. Next, among the remaining players,every player whose district plan does not tie for having the highest sumof scores for items 2+4 is eliminated. Any player not eliminated winsthe game. In the preceding illustrative example, the players' scores foritems 1+3 have the same sum, so the sum of the scores for items 2+4 isthe tiebreaker. The sum of Player 1's scores for items 2+4 is 3. The sumof Player 2's scores for items 2+4 is 5, so Player 2 wins.

Example 10

Strategy games of Example 10 are similar to those of Example 9 and havea taxonomic code A/G/S1/P/2/9/I. They are analog, multi-player gameswith 36 square sectors and a political focus in which two types ofelements—namely two political parties—are present and 9 districts areformed. These games proceed according to simultaneous independent play,so any number of players may play.

In one example, the game is set up exactly as described in Example 9,but the pre-defined goal of each player is to create the most balancedset of political districts. In particular, each player's pre-definedgoal may be to create a district plan that (i) equalizes the number ofdistricts controlled by each party, (ii) equalizes the margin by whicheach party controls its least safe district, and (iii) maximizes thenumber of tied districts that have a voter margin of 0. Item (i) haspriority over (ii), and (ii) has priority over (iii). Each player'sdistrict plan must satisfy requirements A-B (see description of Example9).

At the end of play, each player's final district plan must be displayedand scored. Scoring may include (a) computing the district votermargins, (b) placing scoring tokens appropriately on the scoreboard, and(c) computing the following for each player's final district plan:

-   -   1. The number of districts controlled by the Red Party.    -   2. The lowest voter margin in the districts controlled by the        Red Party.    -   3. The number of districts controlled by the Blue Party.    -   4. The lowest voter margin in the districts controlled by the        Blue Party.    -   5. The magnitude of the difference between items 1 and 3 above        (the “Red-Blue Control Differential”).    -   6. The magnitude of the difference between items 2 and 4 above        (the “Lowest Voter Margin Differential”).    -   7. The number of tied districts (with a voter margin of 0).

The above items may be remembered by placing two red, two blue, andthree gray markers in appropriate places on the rows of each player'sgame board 828.

Identifying the Winner(s)

The winner is identified by process of elimination. First, every playerwhose district plan violates one of the requirements A-B (seedescription of Example 9) is eliminated. Second, among the remainingplayers, every player whose district plan does not tie for having thelowest score for item 5 above is eliminated. Next, among the remainingplayers, every player whose district plan does not tie for having thelowest score for item 6 above is eliminated. Finally, among theremaining players, every player whose district plan does not tie forhaving the highest score for item 7 above is eliminated. Any player whois not eliminated wins the game. If all players' district plans violateone of the requirements A-B (see description of Example 9), all playerslose.

Example of Play

FIGS. 40A and 40B illustrate one possible conclusion of a strategy gameof Example 10, with Player 1's final district plan shown in FIG. 40A andPlayer 2's final district plan shown in FIG. 40B. Bold lines indicatethe districts formed by the layout of expansion markers 808 for eachplayer.

Player 1's scoreboard should show that the voter margin of his/her(Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) Districtis (+19 Blue, +5 Blue, 0, +2 Blue, +12 Red, +8 Blue, +4 Red, +7 Red, +11Red). In player 1's district plan, the Red and Blue Party each control 4districts; the lowest voter margin in the districts controlled by theRed Party is 4; the lowest voter margin in the districts controlled bythe Blue Party is 2; and one district is tied. Player 2's scoreboardshould show that the voter margin of his/her (Brown, Red, Orange,Yellow, Green, Blue, Purple, Pink, Gray) District is (0, +3 Blue, +15Blue, +14 Blue, +3 Red, +25 Red, +15 Blue, +9 Red, +10 Red). In player2's district plan, the Red and Blue Party each control 4 districts; thelowest voter margin in the districts controlled by the Red Party is 3;the lowest voter margin in the districts controlled by the Blue Party is3; and one district is tied.

The scoring of items 1-7 above takes place in rows 1-7 of the game boardand is summarized in the table below. Both players' district planssatisfy requirements A-B, and the players have the same score for item5, so the score for item 6 is the tiebreaker. Player 2 has a lower scorefor item 6, so Player 2 wins the game.

Player 1 Player 2 Item Score Score Item 1 4 4 Item 2 4 3 Item 3 4 4 Item4 2 3 Item 5 0 0 Item 6 2 0 Item 7 1 1

Example 11

Strategy games of Example 11 are generally for 2-4 players and aresimilar to games of Example 3. They have taxonomic codeA/G/S1/P/4/9/U/2-4/EXR. They are analog, multi-player games with 36square sectors in which four types of elements—such as four politicalparties—are present and nine districts are formed. These games proceedaccording to alternating, turn-based play, and moves in categories “E,”“X,” and “R” are allowed.

Game Summary

FIGS. 41-43 illustrate components of a strategy game 900. Gamecomponents not shown in FIGS. 41-43 may be identical to those shown inFIG. 12. In this game, 2-4 players representing opposing politicalparties (Red Lightning Bolts, Orange Suns, Green Diamonds, and BlueMoons) vie for political control of the region 902, which may representa Martian colony, by competitively creating nine political districts outof 36 square sectors 904 in alternating turn-based fashion. At theoutset, the districts are formless and the players know how many voterssupport each party in each community. During the first phase of thegame, players build the political districts by assigning communities topolitical districts one community at a time. They may also reassigncommunities from large districts to adjacent smaller districts in orderto better equalize the district sizes. During the (optional) secondphase of the game, the political status of each district is convertedinto a numerical likelihood of each party winning the district, and anelection is simulated by rolling dice. In the game's final phase, theparties that control the districts are identified, and 3 (1) points areawarded to a party that has sole (joint) control of a district. Theplayer whose party has more points than any other player's party is thewinner. If parties represented by two or more players tie for having themost points, those players jointly win.

FIG. 41 shows examples of two sector tiles 910 that may be used in thisgame. Each sector tile 910 represents a community and has a set ofelements 912 including four icons in its center. Each icon represents anelement, such as one person, or a set number of people, who support apolitical party. There are four non-limiting examples of icons shown,though any suitable icons may be used—a lightning bolt (red), sun(orange), diamond (green), and moon (blue)—corresponding to the fourpolitical parties. The number of identical icons represents the quantity914 of the particular element. For example, the community represented bythe first sector tile 910 on the left of FIG. 41 contains four peopleeach of whom supports a different party. The community represented bythe second sector tile 910 on the right of FIG. 41 contains two peoplewho support the Red Lightning Bolts and two people who support the GreenDiamonds.

FIGS. 42A and 42B provide a visual summary of one possible version ofstrategy game 900. FIG. 42A shows one possible starting position. FIG.42B shows one possible final position for the game, with home basemarkers 906 and expansion markers 908 forming nine districts (indicatedby bold lines). The table below shows the total number of voterssupporting each party in each district of FIG. 42B at the end of thegame. A superscript W indicates that a party wins a district outright. Asuperscript T indicates that a party ties for winning a district.

Red Orange Green Blue Party Party Party Party District Voters VotersVoters Voters Brown 4   5^(W) 4 3 Red 4  5^(T) 2  5^(T) Orange  4^(T) 4^(T)  4^(T)  4^(T) Yellow 3 4 3   6^(W) Green 3 3   4^(W) 2 Blue 3 2 5  6^(W) Purple   6^(W) 5 5 4 Pink 3  5^(T)  5^(T) 3 Gray   6^(W) 3 4 3

The table below shows the total number of districts that each party winsoutright (W) and ties for winning (T). It also shows the total pointsearned by each party assuming that 3 points are earned when a party winsa district outright and 1 point is earned when a party ties for winninga district. In the final tally, the Blue Party wins this game with 8points (2 outright wins+2 ties).

# Outright # (T) Party Wins (W) Ties Points Red 2 1 7 (=3*2 + 1) Orange1 3 6 (=3*1 + 3) Green 1 2 5 (=3*1 + 2) Blue 2 2 8 (=3*2 + 2)

Components

The components for this game are similar to those used in Example 3.They are as follows.

-   -   A region 902, which may be a square in shape and be divided into        36 sector placeholders.    -   36 sector tiles 910 (number of sectors with quantity of each        icon [Red Lightning Bolt, Orange Sun, Green Diamond, Blue Moon]        is in parentheses below)

[1, 1, 1, 1] (2) [2, 1, 1, 0] (1) [1, 2, 1, 0] (1) [1, 1, 2, 0] (1) [2,1, 0, 1] (1) [1, 2, 0, 1] (1) [1, 1, 0, 2] (1) [4, 0, 0, 0] (1) [2, 0,1, 1] (1) [1, 0, 2, 1] (1) [1, 0, 1, 2] (1) [0, 2, 1, 1] (1) [0, 1, 2,1] (1) [0, 1, 1, 2] (1) [0, 4, 0, 0] (1) [2, 2, 0, 0] (1) [2, 0, 2, 0](1) [2, 0, 0, 2] (1) [0, 2, 2, 0] (1) [0, 2, 0, 2] (1) [0, 0, 2, 2] (1)[0, 0, 4, 0] (1) [3, 1, 0, 0] (1) [1, 3, 0, 0] (1) [3, 0, 1, 0] (1) [1,0, 3, 0] (1) [3, 0, 0, 1] (1) [1, 0, 0, 3] (1) [0, 0, 0, 4] (1) [0, 3,1, 0] (1) [0, 1, 3, 0] (1) [0, 3, 0, 1] (1) [0, 1, 0, 3] (1) [0, 0, 3,1] (1) [0, 0, 1, 3] (1)

-   -   9 rectangular home base markers 906 (one for each of 9        districts: Brown, Red, Orange, Yellow, Green, Blue, Purple,        Pink, Gray)    -   162 expansion markers 908 (18 for each of the nine        districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink,        and Gray)    -   36 scoring tokens 606 (nine in each of the colors red, orange,        green, and blue)    -   2 ten-sided dice 618 (1 black, 1 white), each showing the values        0-9    -   Scoreboard 916 (FIG. 43)

Game Setup

The setup is similar that in Example 3. The main difference is that, inthis game, four scoring tokens—one for each color—are stacked on theposition “No. Voters for Each Party=0” in each district's portion of thescoreboard.

Scoreboard and Scoring Tokens

FIG. 43 shows a scoreboard 916 that may be used in strategy game 900.Scoreboard 916 has nine rows 918, one for each district to be formed.During the game, the number of voters that support each political partyin each district is indicated by the position of a scoring token thatmatches the party's color in the row 918 for that district. For example,consider the final position shown in FIG. 42B. In this position, thetotal number of (Red Lightning Bolts, Orange Suns, Green Diamonds, andBlue Moons) in the Brown District is (4, 5, 4, 3) respectively. Toindicate this, a (red, orange, green, blue) scoring token should beplaced at position “No. Voters for Each Party”=(4, 5, 4, 3) respectivelyin the Brown District row 918 of the scoreboard 916. The positions ofthe scoring tokens may be updated after every turn.

Playing the Game

Play may include of the following three phases. The second phase isoptional.

-   -   1. Build political districts    -   2. Run an election (optional)    -   3. Identify the winner

Phase 1: Build Political Districts

This phase proceeds exactly as in Example 3. The only difference is thatthe turns alternate among up to four players instead of two.

Phase 2: Run an Election

Phase 2 is somewhat different in a four-party version of strategy game900 as compared to strategy game 500 in Example 3. In this phase of thegame, the political status in each district is converted into anumerical likelihood of each party winning the district, and an electionin each district is simulated by rolling dice. In particular, thethree-step procedure below (A-B-C) is performed for each districtbeginning with the Brown District.

(A) The table below is used to convert the district's political statusinto a probability of each party winning an election in the district.This is done by (1) ranking the parties according to voter support inthe district (i.e. deciding which party is in 1^(st), 2^(nd), 3^(rd) and4^(th) place); (2) computing the difference in voter support between theparties; and (3) finding the appropriate row in the table below. If twoor more parties have the same voter tally, their ranking isinconsequential but the players must still (arbitrarily) rank them. Forexample, if the political status of the Green district in a four-playergame is (3 Red Voters, 3 Orange Voters, 4 Green Voters, 2 Blue Voters),then the Green (Red, Orange, Blue) Party is in 1^(st) (2^(nd), 3^(1d),4^(th)) place, the difference between 1^(st) and 2^(nd) place is 1voter, the difference between 2^(nd) and 3^(rd) place is 0 voters, andthe difference between 3^(rd) and 4^(th) place is 1 voter. In this case,the Green (Red, Orange, Blue) Party has a 60% (20%, 20%, 0%) chance ofwinning an election in the Green District. In a 2-3 player game, partiesnot represented by an active player are not ignored. These parties canstill win districts and earn points at the end of the game. However,such parties are not allowed to win the game. Only a party representedby an active player may win the game.

Winning Percentage for Party in District Political Status1^(st)/2^(nd)/3^(rd)/4^(th) Place Four parties have same number ofvoters 25/25/25/25 Three parties tied with the most voters 33/33/33/0Two parties tied with the most voters and their 45/45/10/0 voter countexceeds party in 3^(rd) place by 1 Two parties tied with the most votersand their 50/50/0/0 voter count exceeds party in 3^(rd) place by atleast 2 One party is alone in the lead and two parties have 60/20/20/0one less voter than the leader One party is alone in the lead and oneparty has 70/30/0/0 one less voter than the leader One party is alone inthe lead and its voter count 100/0/0/0 exceeds party in 2^(nd) place byat least 2

(B) Next, a random number from 1-100 is produced by simultaneouslyrolling the two 10-sided dice. The value on the black (white) die is thetens (ones) digit of the random number. For example, if the black(white) die shows 7 (1), the result is 71. If the black (white) dieshows 0 (8), the result is 8. The only exception to the above rule isthat a roll of “zero-zero” gives the result of 100. In the unlikelyevent that the result is 100 and three parties are tied for having themost voters in the district, the dice should be re-rolled until a resultbelow 100 is obtained.

(C) The winner of the district's election is then determined bycomparing the random number to the parties' winning percentages in thedistrict. If the random number is less than or equal to the winningpercentage of the 1^(st) place party, the 1^(st) place party wins thedistrict election. Otherwise, if the random number is less than or equalto the sum of the winning percentages of the 1^(st) and 2^(nd) placeparties, the 2^(nd) place party wins the district election. Otherwise,if the random number is less than or equal to the sum of the winningpercentages of the 1^(st), 2^(nd), and 3^(rd) place parties, the 3^(rd)place party wins the district election. If the 1^(st), 2^(nd), and3^(rd) place parties have the same number of voters in a district andthe result is 100, the dice are re-rolled until a value below 100 isobtained. Otherwise, if the random number is greater than the sum of thewinning percentages of the 1^(st), 2^(nd) and 3^(rd) place parties, the4^(th) place party wins the district election. In the example describedin step A, the Green Party wins the Green District election if therandom number is any value from 1-60; the Red Party wins the GreenDistrict election if the random number is any value from 61-80; and theOrange Party wins the Green District election if the random number isany value from 81-100. All scoring tokens are then removed from thatdistrict's portion of the scoreboard, and a single marker matching thecolor of the party that wins the election that is placed on the “RedWins,” “Orange Wins,” “Green Wins,” or “Blue Wins” square in thatdistrict's portion of the scoreboard.

The above procedure is repeated for each of the nine politicaldistricts.

Illustrative Example

We consider the (4-player) game whose final board position is shown inFIG. 42B. The scoreboard for this position is shown in the section “GameSummary” above. Note that a 1^(st) place party automatically wins adistrict election (with 100% chance) if it has at least two more votersthan the 2^(nd) place party. Thus, in the game at hand, the Blue Partyautomatically wins the Yellow District, and the Red Party automaticallywins the Gray District.

Dice 618 are then rolled to determine the winners of the seven districtsin which there is

not an automatic winner. The election results are summarized in thetable below. In the table, the dice rolls are shown in the “Dice Roll”column, and “R,” “O,” “G,” and “B” refer to the Red, Orange, Green, andBlue Party respectively. The overall result is that the Blue (Red,Green) Party wins the elections in three (two, four) districts.

Red Orange Green Blue Party Winning Party Party Party Party RankingPercentage Dice Election District Voters Voters Voters Voters (1^(st),2^(nd), 3^(rd), 4^(th)) 1^(st)/2^(nd)/3^(rd)/4^(th) Roll Result Brown 45 4 3 O, R, G, B 60/20/20/0 81 Green Wins Red 4 5 2 5 O, B, R, G45/45/10/0 45 Orange Wins Orange 4 4 4 4 R, O, G, B 25/25/25/25 34Orange Wins Yellow 3 4 3 6 B, O, R, G 100/0/0/0 — Blue Wins Green 3 3 42 G, R, O, B 60/20/20/0  2 Green Wins Blue 3 2 5 6 B, G, R, O 70/30/0/0100  Green Wins Purple 6 5 5 4 R, O, G, B 60/20/20/0 57 Red Wins Pink 35 5 3 O, G, R, B 50/50/0/0 66 Green Wins Gray 6 3 4 3 R, G, O, B100/0/0/0 — Red Wins

Phase 3: Identify the Winner(s)

In the game's final phase, the overall winner is identified. If phase 2was played, the winner is the player whose party wins the most districtelections. If more than one party ties for winning the most districtelections, these parties together win the game and the result is a tie.For example, the Green Party wins the game shown in the table above.

If phase 2 was not played, the scoreboard is used to identify the partythat controls each district. Two or more parties jointly control adistrict if they tie for having the greatest number of voters in thedistrict. If there is no tie, the party with the greatest number ofvoters in the district solely controls the district. Each party receivesthree points for each district that it solely controls and one point foreach district that it jointly controls. The player whose party has morepoints than any other player's party is the winner. If partiesrepresented by two or more players tie for having the most points, thoseplayers jointly win. A party not represented by a player may not win thegame. Refer to section “Game Summary” above to see who wins the gamewhose final position is shown in FIG. 42B if phase 2 is not played.

Example 12

This example encompasses several games that are larger versions of thegame described in Example 11. These larger versions are played with moresectors than Example 11 but are otherwise very similar to Example 11.The relationship of these games to Example 11 is analogous to therelationship of Examples 4-8 to Example 3.

The taxonomic codes for five possible games included in this example arelisted below. All games are analog, multi-player games with squaresectors and a political focus in which four types of elements—namelyfour political parties—are present, the game proceeds according toalternating turn-based play, there are 2-4 players, and moves incategories “E,” “X,” and “R” are allowed.

-   -   A/G/S2/P/4/12/U/2-4/EXR    -   A/G/S3/P/4/15/U/2-4/EXR    -   A/G/S4/P/4/15/U/2-4/EXR    -   A/G/S5/P/4/21/U/2-4/EXR    -   A/G/S6/P/4/21/U/2-4/EXR

The aforementioned five games get progressively larger with 72, 90, 121,169, and 210 sectors respectively.

The move types allowed in the aforementioned five games are identical tothe move types described in Examples 4-8 respectively.

The main difference between these games and Examples 4-8 is that up tofour players can play these games.

Play of any of the above games proceeds in a manner similar to Example11. During the first phase of the game, players build the politicaldistricts by assigning sectors to political districts one sector at atime. They may also reassign sectors from large districts to adjacentsmaller districts in order to better equalize the district sizes. Duringthe (optional) second phase of the game, the political status of eachdistrict is converted into a numerical likelihood of each party winningthe district, and an election is simulated by rolling dice. In thegame's final phase, the parties that control the districts areidentified, and 3 (1) points are awarded to a party that has sole(joint) control of a district. The player whose party has more pointsthan any other player's party is the winner. If parties represented bytwo or more players tie for having the most points, those playersjointly win.

Example 13

Strategy games of Example 13 are a combination of the simultaneousindependent play undertaken in Example 9 and the four-party environmentconsidered in Example 11. Such games have taxonomic code A/G/S1/P/4/9/I.They are analog, multi-player games with 36 square sectors and apolitical focus in which four types of elements—namely four politicalparties—are present and nine districts are formed. The games proceedaccording to simultaneous independent play, so any number of players mayparticipate.

Components

In one example, the game components are highly similar to the componentsused in Examples 9 and 11 and shown in FIGS. 36-37 and 41-43. A timer814 may be needed to play this game, if time constraints are being used.In addition, each player should have a copy of the same game set whichcontains the following items:

-   -   36 sector tiles 910 having the same markings as the sectors in        Example 11    -   80 expansion markers 908 (8 Brown, 10 Red, 10 Orange, 8 Yellow,        10 Green, 10 Blue, 8 Purple, 8 Pink, 8 Gray)    -   36 scoring tokens 812 (4 for each of the nine district colors)    -   A scoreboard 916 (same as in FIG. 43)    -   A game board 818 (similar to FIG. 37 but with more scoring rows        on the left side)        Setup (about 10 minutes)

The setup is very similar to that in Example 9, but with each playerlaying out a region 902 having sectors 904. Overall, each player createsa copy of the exact same 6×6 sector arrangement in his/her playingregion 902 and organizes piles of markers within his/her playing area toprepare for what follows.

Playing the Game

Play may include four phases, each having a different pre-defined goal,as listed below. Phases 2-4 are optional.

-   -   1. Build political districts that maximize the Red Party's        advantage    -   2. Build political districts that maximize the Orange Party's        advantage    -   3. Build political districts that maximize the Green Party's        advantage    -   4. Build political districts that maximize the Blue Party's        advantage

Each phase proceeds like a phase described in Example 9. Players usetheir markers, scoreboard, and game board as desired to try to achievethe pre-defined goal. The main goal in each phase is to create 9political districts—i.e. a district plan—in which the concerned partycontrols as many districts as possible. A party controls a district ifit has strictly more voters in a district than any other party. Eachplayer's secondary goal is to maximize the total amount—summed over thedistricts controlled by the concerned party—by which the concerned partyleads its closest adversary in the districts that it controls.

At the end of each phase of the game, each player tracks his/her scorewith respect to the pre-defined goals above by placing markers onsquares in relevant rows 828 of his/her game board 818. Penalties areassessed if a player's district plan violates requirement A or B (seeExample 9).

Identifying the Winner(s)

The winner of the game is identified, by process of elimination, bylooking at the markers on the rows 828 of each player's game board 818.These markers show the scores for up to eight items:

-   -   1. Number of districts controlled by the Red Party in phase 1    -   2. Total amount by which the Red Party controls its districts in        phase 1    -   3. Number of districts controlled by the Orange Party in phase 2        (if phase 2 played)    -   4. Total amount by which the Orange Party controls its districts        in phase 2 (if played)    -   5. Number of districts controlled by the Green Party in phase 3        (if phase 3 played)    -   6. Total amount by which the Green Party controls its districts        in phase 3 (if played)    -   7. Number of districts controlled by the Blue Party in phase 4        (if phase 4 played)    -   8. Total amount by which the Blue Party controls its districts        in phase 4 (if played)

If all phases are played, the winner is identified as follows. First,every player whose district plan does not tie for having the highest sumof scores for items 1+3+5+7 is eliminated. Next, among the remainingplayers, every player whose district plan does not tie for having thehighest sum of scores for items 2+4+6+8 is eliminated. Any player noteliminated wins the game.

Example 14

Strategy games of Example 14 combine the (optionally) time-limited,simultaneous independent play undertaken in Example 10 and thefour-party environment considered in Example 11. Such games have ataxonomic code A/G/S1/P/4/9/I. They are analog, multi-player games with36 square sectors and a political focus in which four types ofelements—namely four political parties—are present and nine districtsare formed. These games proceed according to simultaneous independentplay, so any number of players may participate.

Components

In one example, the game components are nearly identical to those inExample 13. The only difference is that a few additional markers areneeded to track the final score on the game board.

Setup (about 10 Minutes)

The setup is very similar to that in Example 9. Overall, each playercreates a copy of the exact same 6×6 sector arrangement in his/herregion.

Scoreboard and Scoring Tokens

The scoreboard and scoring tokens may be the same as in Example 11.

Playing the Game

Play proceeds just as in Example 10. As long as time has not expired,players may use their markers, scoreboard, and game board as desired totry to achieve the desired goal. Each player's main goal is to create 9political districts—i.e. a district plan—in which all four partiescontrol the same number of districts. A party controls a district if ithas strictly more voters in a district than any other party. Eachplayer's secondary goal is to equalize the total amount—summed over thedistricts controlled by each party—by which each party leads its closestadversary in the districts that it controls. Each player's tertiary goalis to maximize the number of districts in which all four parties havethe same number of voters.

At the end of play (e.g., when time expires), each player tracks his/herscore with respect to the 3 goals above by placing markers on squares inthe left part of his/her game board. Penalties are assessed if aplayer's district plan violates requirement A or B (see Example 9).

Identifying the Winner(s)

The winner of the game is identified by looking at the markers on theleft side of each player's game board. These markers show the scores foreleven items:

-   -   1. Number of districts controlled by the Red Party    -   2. Total amount by which the Red Party controls its districts    -   3. Number of districts controlled by the Orange Party    -   4. Total amount by which the Orange Party controls its districts    -   5. Number of districts controlled by the Green Party    -   6. Total amount by which the Green Party controls its districts    -   7. Number of districts controlled by the Blue Party    -   8. Total amount by which the Blue Party controls its districts    -   9. The magnitude of the difference between the highest value        among items 1, 3, 5, and 7 and the lowest value among items 1,        3, 5, and 7.    -   10. The magnitude of the difference between the highest value        among items 2, 4, 6, and 8 and the lowest value among items 2,        4, 6, and 8.    -   11. The number of districts in which all parties have the same        number of voters.

The winner is identified by process of elimination. First, every playerwhose district plan violates one of the requirements A-B (seedescription of Example 9) is eliminated. Second, among the remainingplayers, every player whose district plan does not tie for having thelowest score for item 9 above is eliminated. Next, among the remainingplayers, every player whose district plan does not tie for having thelowest score for item 10 above is eliminated. Finally, among theremaining players, every player whose district plan does not tie forhaving the highest score for item 11 above is eliminated. Any player whois not eliminated wins the game. If all players' district plans violateone of the requirements A-B (see description of Example 9), all playerslose.

Example 15

Strategy games of Example 15 have taxonomic code A/G/T1/P/2/9/U/2/EXRand are triangular-sector games similar to the square-sector games ofExample 3. They are analog, multi-player games with 54 sectors in theshape of an equilateral triangle. They have two types of elements—suchas two political parties—and nine districts are formed. The gamesproceed according to alternating, turn-based play; there are twoplayers; and moves in categories “E,” “X,” and “R” are allowed.

Game Summary

FIGS. 44-46 illustrate one example of a strategy game 1600. In thisgame, two players representing opposing political parties—Red andBlue—vie for political control of a hexagonal region 1602 bycompetitively creating nine political districts out of 54 triangularsectors in alternating, turn-based fashion. The region 1602 may bedivided into 54 sector placeholders 1608, and sector tiles 1606 may belaid out, on each sector placeholder 1608, to form sectors 1604 as partof the game set-up. Each sector 1604 may represent a community, and theset of elements on the sector tile 1606 for each sector 1604 includes atype (e.g., favored political party) and quantity (e.g., voter margin bywhich the indicated party is favored). The strategy game 1600 may alsoinclude a set of markers 1610, divided into marker subsets 1612 each ofwhich represents a district. Each marker subset may include a home basemarker 1614 and at least one expansion marker 1616. The number of markersubsets 1612 preferably equals the number of districts to be formedduring the game. The total number of home base markers 1614 andexpansion markers 1616 for each marker subset 1612 should be sufficientto form districts of appropriate size for the game. The strategy game1600 may further include a scoreboard 1618 (same as FIG. 13), which mayhave one row 1624 for each district to be formed, and one scoring token1620 for each district to be formed. The strategy game 1600 may alsoinclude two dice 1622, which may be ten sided dice, each having adifferent color such as one black and one white.

During the first phase of strategy game 1600, players build thepolitical districts by assigning sectors to political districts onesector at a time. They may also reassign sectors from large districts toadjacent smaller districts in order to better equalize the districtsizes. During the optional second phase of strategy game 1600, thepolitical margin in each district is converted into a numericallikelihood of each party winning the district, and an election issimulated by rolling dice 1622. The winner is the player whose partycontrols more districts than his/her opponent. A tie is possible ifplayers skip phase 2 of the game.

FIG. 45 shows one possible initial position for a strategy game 1600, inwhich the sector tiles 1606 have been placed on the region, with one oneach sector placeholder 1608, to form sectors 1604. FIG. 46 shows onepossible final position for a game having the initial position shown inFIG. 45, with the home base markers 1614 and expansion markers 1616placed on the region 1602, one per sector 1604, to form nine districts(indicated by bold lines). The table below shows the final result forthe final position shown in FIG. 46. If phase 2 is not played, the RedParty wins the game by a score of 5 districts to 3 districts (with onetied district):

Party in By How District Control Much Brown Blue 12 Red Red 8 OrangeNeither 0 Yellow Red 2 Green Red 1 Blue Blue 11 Purple Blue 8 Pink Red 6Gray Red 14

Game Components

The game components are as follows:

-   -   54 triangular sector tiles 1606 (quantity of each sector tile        type is shown in parentheses below):

+9 Red (3) +5 Red (3) +1 Red (2) +3 Blue (3) +7 Blue (3) +8 Red (3) +4Red (3) 0 (2) +4 Blue (3) +8 Blue (3) +7 Red (3) +3 Red (3) +1 Blue (2)+5 Blue (3) +9 Blue (3) +6 Red (3) +2 Red (3) +2 Blue (3) +6 Blue (3)

-   -   9 home base markers 1614 (one for each of 9 districts: Brown,        Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray)    -   180 expansion markers 1616 (20 for each of the nine        districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink,        and Gray)    -   9 scoring tokens 1620 (one for each of 9 districts—Brown, Red,        Orange, Yellow, Green, Blue, Purple, Pink, and Gray), each being        cube-shaped with faces showing the numbers 10, 20, 30, 40, and        50    -   2 ten-sided dice 1622 (1 black, 1 white), each showing the        values 0-9    -   A scoreboard 1618

Playing the Game

The game has three phases and plays in a manner similar to Example 3.

In phase 1, players take alternating turns beginning with the startingplayer. During a player's turn, he/she (A) makes one move and then (B)records the move on two scoreboards. Forfeiting a turn (i.e. passing ona turn) is not allowed. All moves must be of type 1, 1A, 2, 2A, 3, or 3Abelow. About 60 moves—30 by each player—are made in phase 1.

Phase 1 is divided into three stages. In stage 1, only moves of type 1,2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1exist and (ii) fewer than 9 districts have been established. In stage 2,only moves of type 1A, 2, and 3 are allowed. Play enters stage 3immediately after the 9^(th) district is established. In stage 3, thenext move must be of type 2 or 3A if a move of type 2 exists. Otherwise,the next move must be of type 2A or 3A. Phase 1 concludes when no legalmoves exist.

The rules may allow districts to be formed by the following six types ofmoves. The meanings of the phrases “connectedness,” “captured tile,” and“trapped district” are analogous to those in Example 3.

-   -   1 Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (a′) The        sector must be at least three steps away from all previously        placed home base markers. (b′) There must be space to grow this        district to size 6 or more. Sectors 1 and 7 in FIG. 44 are five        steps away from each other.    -   1A Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (c′) The        sector must be in the largest open space on the board. An open        space is a set of connected vacant sectors. (d′) Among the        sectors satisfying this criterion, the sector must be the        farthest, in number of steps, from a previously placed home base        marker.    -   2 Expand a district by placing one of its expansion markers on a        vacant sector. This move must meet three requirements. (e′) The        district must remain connected. (f) The district's new        size—including this new sector and any sectors that are        captured—must not be greater than 6 sectors. (g′) No district        may be trapped.    -   2A Expand a district by placing one of its expansion markers on        a vacant sector. This move must meet three requirements. (e′)        The district must remain connected. (h′) Only the smallest        expandable district may be expanded. (i′) No sectors may be        captured.    -   3 (Same as in Examples 3-8 and 11-12)    -   3A (Same as in Examples 3-8 and 11-12)

Phases 2-3 are played almost exactly as in Example 3.

Rules for a Symmetric Game

A symmetric version of this game may be played if players are concernedabout bias in the initial sector arrangement. The purpose of a symmetricgame is to remove bias from the initial sector arrangement and therebygive each party—Red and Blue—a fair chance of winning the game.

A symmetric game has three additional rules compared to a non-symmetricgame. Rule 1 guarantees a symmetrical initial sector arrangement,whereas rules 2 and 3 minimize the possibility of a symmetric boardposition during play. The three rules are as follows.

-   -   1. During the game setup, the sector tile arrangement must be        counter-symmetric with respect to the dot in the center of the        region. A counter-symmetric sector arrangement is one in which        the two sectors comprising every pair of diametrically-opposed        sectors—i.e. every two sectors that are on exact opposite sides        of the region—have opposite voter margins—for example “+4 Red”        and “+4 Blue.” The procedure may be nearly identical to that        described in the section “Rules for a symmetric game” in Example        3.    -   2. During the second and third moves of phase 1 (i.e. the 1^(st)        move made by the player who goes second, and the 2^(nd) move        made by the player who goes first), no marker may be placed on        sector that is diametrically opposed to a sector on which a        marker has already been placed.    -   3. During phase 1, any move that would result in a district        that (a) has size six and (b) coincides with the hexagon in the        center of the region is strictly forbidden.

Example 16

Strategy games of Example 16 are larger versions of the games describedin Example 15. These games have taxonomic code A/G/T2/P/2/12/U/2/EXR.They are analog, multi-player games with 96 triangular sectors in which12 districts are formed. Such games proceed according to alternating,turn-based play; there are two players; and moves in categories “E,”“X,” and “R” are allowed.

In one example, a strategy game may be very similar to Example 15. Insuch a game, two players—Red and Blue—vie for political control of thehexagonal region by competitively creating 12 political districts (whoseaverage size is 8) out of 96 triangular sectors. The sectors may bepre-established on the region, or formed by placing one sector tile oneach of 96 sector placeholders on the region. The game can be playedwith any 96 triangular sectors, though it is preferred that the sets ofsectors favoring red and blue be identical—for example 6 each of sectortiles “+2 Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (96 sectorstotal). Players take alternating turns beginning with the startingplayer. During a player's turn, he/she (A) makes one move and then (B)records the move on two scoreboards. All moves must be of type 1, 1A, 2,2A, 3, or 3A below. About 100 moves-50 by each player—are made in agame. The game ends when no legal moves exist. The winner is the playerwhose party controls more districts than his/her opponent.

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than 12 districts have been established. In stage 2, onlymoves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediatelyafter the 12^(th) district is established. In stage 3, the next movemust be of type 2 or 3A if a move of type 2 exists. Otherwise, the nextmove must be of type 2A or 3A. Play concludes when no legal moves exist.

The six types of legal moves are summarized below.

-   -   1 Establish a district on a sector that is at least 3 steps away        from all other home base markers. There must be space to grow        this district to size 8 or more.    -   1A Establish a district on a sector that ties for being the        farthest, in number of steps, from a home base marker (among the        sectors in the largest open space on the board).    -   2 Expand a district so (e′) it remains connected, (f) its new        size (including any captured sectors) is 8 or less, and (g′) no        district is trapped.    -   2A Expand the smallest expandable district so (e′) it remains        connected, (i′) no sectors are captured, and (o′) its new size        is 21 or less.    -   3 (Same as in Examples 3-8, 11-12, and 15)    -   3A (Same as in Examples 3-8, 11-12, and 15)

Example 17

Strategy games of Example 17 are even larger versions of the gamesdescribed in Example 15. These games have taxonomic codeA/G/T3/P/2/15/U/2/EXR. They are analog, multi-player games with 150triangular sectors, and may have a political focus in which two types ofelements—namely two political parties—are present and 15 districts areformed. These games may proceed according to alternating, turn-basedplay; may have two players; and moves in categories “E,” “X,” and “R”may be allowed.

Game Summary

FIGS. 47-48 illustrate elements of a strategy game 1000. In this game,two players representing opposing political parties—Red and Blue—vie forpolitical control of a hexagonal region 1002 by competitively creatingfifteen political districts out of 150 triangular sectors 1004 inalternating, turn-based fashion. The region 1002 may be divided into 150sector placeholders, and sector tiles 1006 may be laid out, on eachsector placeholder, to form sectors 1004 as part of the game set-up.Alternatively, each sector 1004 may be pre-established on the region1002. Each sector 1004 may represent a community, and the set ofelements 1008 in each sector 1004 includes a type (e.g., favoredpolitical party) and quantity (e.g., voter margin by which the indicatedparty if favored). The strategy game 1000 may also include a set ofmarkers, divided into marker subsets that each represents a district.Each marker subset may include a home base marker 1010 and at least oneexpansion marker 1012. The number of marker subsets preferably equalsthe number of districts to be formed during the game. The total numberof home base markers 1010 and expansion markers 1012 for each markersubset should be sufficient to form districts of appropriate size forthe game. The strategy game 1000 may further include a scoreboard 1014,which may have one row 1016 for each district to be formed, and scoringtokens sufficient to track the score during play. The strategy game 1000may also include two dice (which may be identical to dice 1622).

During the first phase of the game, players build the politicaldistricts by assigning sectors 1004 to political districts, one sectorper turn. They may also reassign sectors during a turn from largedistricts to small districts in order to better equalize the districtsizes. During the (optional) second phase of the game, the politicalmargin in each district is converted into a numerical likelihood of eachparty winning the district, and an election is simulated by rollingdice. The winner is the player whose party controls more districts thanhis/her opponent. A tie is possible if players skip the game's secondphase.

Game Components

The game components may be as follows:

-   -   150 triangular sector tiles 1006 (quantity of each sector type        is shown in parentheses below):

+9 Red (8) +5 Red (8) +1 Red (8) +3 Blue (8) +7 Blue (8) +8 Red (8) +4Red (8) 0 (6) +4 Blue (8) +8 Blue (8) +7 Red (8) +3 Red (8) +1 Blue (8)+5 Blue (8) +9 Blue (8) +6 Red (8) +2 Red (8) +2 Blue (8) +6 Blue (8)

-   -   15 home base markers 1010 (one for each of the 15        districts—Brown, Red, Orange, Yellow, Light Green, Dark Green,        Light Blue, Dark Blue, Purple, Pink, Light Gray, Dark Gray,        Black, White, Gold)    -   450 expansion markers 1012 (30 for each of the aforementioned 15        districts)    -   60 scoring tokens (four for each of the aforementioned 15        districts)    -   2 ten-sided dice (1 black, 1 white), each showing the values 0-9    -   Scoreboard 1014 (FIG. 48)

Playing the Game

The play is very similar to Example 15 but is more challenging owing tothe many sectors. Play may include the following three phases. Thesecond phase is optional.

-   -   1. Build political districts    -   2. Run an election (optional)    -   3. Identify the winner

Phase 1: Build Political Districts Summary

During the first phase of strategy game 1000, players take turnsassigning/reassigning sectors to political districts, one sector at atime, until every sector belongs to a political district. The assignmentof a sector 1004 to a political district is accomplished by placing ahome base marker 1010 or expansion marker 1012 on a sector tile.Reassignment of a sector 1004 from a district to another district may beaccomplished by changing the color of the marker on a sector tile. Atthe end of the first phase, 15 political districts (e.g., Brown, Red,Orange, Yellow, Light Green, Dark Green, Light Blue, Dark Blue, Purple,Pink, Light Gray, Dark Gray, Black, White, and Gold) will be formed onthe region 1002.

Each district evolves in the same general way. Initially, it isformless. At some point, it is established when its home base marker1010 is placed on a vacant sector 1004. It is then expanded whenever oneof its expansion markers 1012 is placed on a vacant sector that isadjacent to a sector that already belongs to the district. Later, it maybe resized so its size is more similar to neighboring districts byreassignment of sectors 1004.

The process of building political districts is relatively unrestricted.There is no general requirement for the sequence in which, or locationswhere, districts are constructed. Once begun, the construction of adistrict may be temporarily halted while players take turnsestablishing, expanding, and/or resizing other districts. There may notbe any district size requirement. However, the rules may encourage orrequire the creation of districts of size 10 (meaning that each districtis formed from 10 sectors 1004).

Importantly, all marker subsets and all sectors are available to allplayers. No player “owns” any marker subset or sector. As long as therules below are followed, any player may contribute to building anydistrict during any turn. No matter which player established a district,any other player may expand the district or reassign a sector from thatdistrict to another district.

FIG. 47 illustrates one possible final position at the conclusion ofgame play, with fifteen districts formed by the home base markers 1010and expansion markers 1012 (the boundaries of which are indicated bybold lines).

Details

Players take turns beginning with the starting player. During a player'sturn, he/she (A) makes one move and then (B) records the move on thescoreboard.

All moves must be of type 1, 1A, 2, 2A, 3, or 3A. Moves of type 1 and 1Aestablish a new district. Moves of type 2 and 2A expand an existing(i.e. already established) district. Moves of type 3 and 3A resize twoadjacent districts. “A” means “alternate move.”

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than 15 districts have been established. In stage 2, onlymoves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediatelyafter the 15^(th) district is established. In stage 3, the next movemust be of type 2 or 3A if a move of type 2 exists. Otherwise, the nextmove must be of type 2A or 3A. In many games, stage 2 is skipped andplay proceeds directly from stage 1 to stage 3. Phase 1 concludes whenno legal moves exist.

The six types of legal moves are as follows. The meanings of the phrases“connectedness,” “captured tile,” and “trapped district” are analogousto those in Example 3.

-   -   1 Establish a district on a sector that is at least 4 steps away        from all other home base markers. There must be space to grow        this district to size 10 or more connected sectors.    -   1A Establish a district on a sector that ties for being the        farthest, in number of steps, from a home base marker (among the        sectors in the largest open space on the board).    -   2 Expand a district so (e′) it remains connected, (f) its new        size (including any captured sectors) is 10 or less, and (g′) no        district is trapped.    -   2A Expand the smallest expandable district so (e′) it remains        connected, (i′) no sectors are captured, and (o′) its new size        is 31 or less.    -   3 Reassign a sector from one district to another. (Same as        Examples 3-8, 11-12, 15-16)    -   3A Reassign a sector from one district to another. (Same as        Examples 3-8, 11-12, 15-16)

Phase 2: Run an Election

The second phase of strategy game 1000 is optional. In the second phase,the political margin in each district is converted into a numericallikelihood of each party winning the district, and an election in eachdistrict is simulated by rolling the dice.

Each district shown in the final position (e.g., in FIG. 47) isindividually considered. First, using the table below, the politicalmargin for the party with more voters in the district is converted intoa numerical likelihood of that party winning an election in thedistrict.

Political Winning Margin Likelihood 0 50% +1 59% +2 67% +3 74% +4 80% +585% +6 89% +7 92% +8 94% +9 96% +10 98% +11 99% +12 or more 100% 

Next, a random number from 1-100 is produced by simultaneously rollingthe two 10-sided dice. See the section “Phase 2: Run an election” inExample 3 for more details.

The random number is then compared to the winning percentage. If therandom number is less than or equal to the winning percentage, the partywith more voters in the district wins the district election. If therandom number is greater than the winning percentage, the party withfewer voters in the district wins the district election. If both partieshave a 50% chance of winning, the Blue Party wins if the random numberis 1-50 and the Red Party wins if the random number is 51-100. After thewinner of an election is identified, the unused scoring token thatmatches the district color is placed on the “Blue Wins” or “Red Wins”square in that district's row in the scoreboard.

The above procedure is repeated for each of the 15 political districts.

Phase 3: Identify the Winner

In the game's final phase, the winner is identified. If phase 2 isplayed, the winner is the player whose party wins eight or more districtelections.

If phase 2 is not played, the scoreboard is used to identify the partythat controls each district, i.e. the party with more voters in eachdistrict. The winner is the player whose party controls more districtsthan his/her opponent. If the two players control an equal number ofdistricts, the result is a tie.

Rules for a Symmetric Game

As with many previous examples, strategy game 1000 can be played as asymmetric game. The purpose of a symmetric game is to remove bias fromthe initial sector arrangement and give each party—Red and Blue—a fairchance of winning the game.

A symmetric game for strategy game 1000 has three additional rulescompared to a regular game. Rule 1 creates a symmetric initial sectorarrangement, and rules 2 and 3 reduce the possibility of a symmetricposition during play. The three rules are as follows.

-   -   1. During the game setup, the sector arrangement must be        counter-symmetric with respect to an imaginary dot in the center        of the region. The procedure for doing this may be nearly        identical to that described with respect to Example 3.    -   2. During the 2^(nd), 3^(rd), 4^(th), and 5^(th) moves of phase        1 (i.e. the 1^(st) and 2^(nd) moves made by the player who goes        second, and the 2^(nd) and 3^(rd) moves made by the player who        goes first), no marker may be placed on a sector that is        diametrically opposed to a sector on which a marker has already        been placed.    -   3. During phase 1, no move may result in a district that (a) has        size ten, (b) has a political margin of zero, and (c) entirely        covers the small hexagon at the center of the region 1002.

Example 18

Strategy games of this Example 18 are even larger versions of the gameswith hexagonal regions and triangular sectors described in Examples15-17. These games have taxonomic code A/G/T4/P/2/27/U/2/EXR. They areanalog, multi-player games with 216 triangular sectors, and may have apolitical focus in which two types of elements—namely two politicalparties—are present and 27 districts are formed. These games proceedaccording to alternating, turn-based play; there are two players; andmoves in categories “E,” “X,” and “R” are allowed.

In one example, two players—Red and Blue—vie for political control ofgiant hexagonal region by competitively creating 27 political districts(whose average size is 8) out of 216 triangular communities. The gamecan be played with any 216 triangular sectors in which the sets of redand blue sectors are identical—for example 12 each of sector tiles “+2Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (192 sector tiles); 8 eachof sector tiles “+1 Red” and “+1 Blue” (16 sector tiles); and 8 sectortiles with a voter margin of 0. Players take turns beginning with thestarting player. During a player's turn, he/she (A) makes one move andthen (B) records the move on three scoreboards. All moves must be oftype 1, 1A, 2, 2A, 3, or 3A below. About 240 moves—120 by eachplayer—are made in a game. The game ends when no legal moves exist. Thewinner is the player whose party controls more districts than his/heropponent.

Play is divided into three stages. In stage 1, only moves of type 1, 2,and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 existand (ii) fewer than 27 districts have been established. In stage 2, onlymoves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediatelyafter the 27^(th) district is established. In stage 3, the next movemust be of type 2 or 3A if a move of type 2 exists. Otherwise, the nextmove must be of type 2A or 3A. Play concludes when no legal moves exist.

This game has the same sector shape and same final average district sizeas Example 16. Thus, the details for the six types of legal moves—listedbelow—are identical to Example 16.

-   -   1 Establish a district on a sector that is at least 3 steps away        from all other home base markers. There must be space to grow        this district to size 8. (Same as Example 16)    -   1A Establish a district on a sector that ties for being the        farthest, in number of steps, from a home base marker (among the        sectors in the largest open space on the board). (Same as        Example 16)    -   2 Expand a district so (e′) it remains connected, (f) its new        size (including any captured sectors) is 8 or less, and (g′) no        district is trapped. (Same as Example 16)    -   2A Expand the smallest expandable district so (e′) it remains        connected, (i′) no sectors are captured, and (o′) its new size        is 21 or less. (Same as Example 16)    -   3 (Same as Examples 3-8, 11-12, 15-17)    -   3A (Same as Examples 3-8, 11-12, 15-17)

Example 19

Strategy games of Example 19 combine the simultaneous independent playundertaken in Example 9 and the triangular sectors used in Example 15.These games have taxonomic code A/G/T1/P/2/9/I. They are analog,multi-player games with 54 triangular sectors and a political focus inwhich two types of elements (e.g., two political parties) are presentand nine districts are formed. The game paradigm is simultaneousindependent play, so any number of players may participate.

Components

The game components are highly similar to the components used in Example15, shown in FIGS. 44-46, with the addition of a game board 1100 asshown in FIG. 49, and a timer (such as timer 814 of Example 8) if timeconstraints will be used. Accordingly each player should have a copy ofthe same game set which contains the following items:

-   -   Region 1602    -   54 sector tiles 1606 having the same markings as the sectors in        Example 15    -   112 expansion markers 1616 (12 Brown, 14 Red, 12 Orange, 12        Yellow, 12 Green, 14 Blue, 12 Purple, 12 Pink, 12 Gray). These        are simply referred to as “markers.”    -   9 scoring tokens (same as in Example 15)    -   Scoreboard 1618 (same as FIG. 13)    -   Game board 1100, which may have a duplicate region 1102 divided        into 54 duplicate sector placeholders 1106. The game board 1100        may also have a plurality of rows 1104 configured to allow a        player to track aspects of districts formed on the game board        1100.

Playing the Game

Play may include of the following two phases. Phase 2 is optional.

-   -   1. Build political districts that maximize the Red Party's        advantage    -   2. Build political districts that maximize the Blue Party's        advantage

Each phase proceeds like a phase described in Example 9. Players usetheir markers, scoreboard, and game board as desired to try to achievethe desired goal. The main goal in each phase is to create 9 politicaldistricts of equal size—i.e. a district plan—in which the concernedparty controls as many districts as possible. Each player's secondarygoal is to make the “voter margin in the district that the concernedparty controls by the least amount” as high as possible. In the districtplan, (A) each sector 1604 must be assigned to exactly one district and(B) each district must consist of six connected tiles as in FIG. 7.

At the end of each phase of the game, each player tracks his/her scorewith respect to the two goals above by placing markers on the relevantsquare of the relevant row 1104 of his/her game board 1100. Penaltiesare assessed if a player's district plan violates requirement A or Babove.

Identifying the Winner(s)

The winner of the game is identified, by process of elimination, bylooking at the markers on the left side of each player's game board.These markers show the scores for up to four items:

-   -   1. Number of districts controlled by the Red Party in phase 1    -   2. Lowest voter margin in the districts controlled by the Red        Party in phase 1    -   3. Number of districts controlled by the Blue Party in phase 2        (if phase 2 played)    -   4. Lowest voter margin in the districts controlled by the Blue        Party in phase 2 (if played)

If phase 2 is not played, the winner is identified as follows. First,every player whose district plan does not tie for having the highestscore for item 1 above is eliminated. Next, among the remaining players,every player whose district plan does not tie for having the highestscore for item 2 above is eliminated. Any player who is not eliminatedwins the game.

If phase 2 is played, the winner is identified as follows. First, everyplayer whose district plan does not tie for having the highest sum ofscores for items 1+3 is eliminated. Next, among the remaining players,every player whose district plan does not tie for having the highest sumof scores for items 2+4 is eliminated. Any player not eliminated winsthe game.

Example 20

Strategy games of Example 20 combine the simultaneous independent playundertaken in Example 10 and the triangular sectors used in Example 15.These games have taxonomic code A/G/T1/P/2/9/I. They are analog,multi-player games with 54 triangular sectors and a political focus inwhich two types of elements—two political parties—are present and 9districts are formed. The playing paradigm is simultaneous independentplay, so any number of players may play.

Game Summary

In one example, the game is set up exactly as described in Example 19,but the pre-defined goal of each player is to create the most balancedset of political districts. The region has an American-style, two-partypolitical system in which one person is elected to represent eachpolitical district. At the outset, the districts are formless and theplayers know the political status of each sector (i.e. which party itscitizens favor and by how much). During the game, players simultaneouslyand independently work on identical copies of the region map to createpolitical districts that equalize the political advantage of the twoparties, Red and Blue. The winner is the player who creates the mostbalanced set of political districts.

The game components, setup, sectors, scoreboard, and scoring methods areidentical to Example 19 except that each player uses an additional 3gray markers to score three additional items on his/her game board.

Playing the Game

Play proceeds as in Example 19, except with the pre-defined goal tocreate a district plan that (i) equalizes the number of districtscontrolled by each party, (ii) equalizes the margin by which each partycontrols its least safe district, and (iii) maximizes the number of tieddistricts that have a voter margin of 0. Item (i) has priority over(ii), and (ii) has priority over (iii). Each player's district plan mustsatisfy requirements A-B as stated in the section “Playing the game” inthe description of Example 19.

At the conclusion of play, the players (a) compute the district votermargins, (b) place scoring tokens appropriately on the scoreboard, and(c) compute the following for each player's final district plan:

-   -   1. The number of districts controlled by the Red Party.    -   2. The lowest voter margin in the districts controlled by the        Red Party.    -   3. The number of districts controlled by the Blue Party.    -   4. The lowest voter margin in the districts controlled by the        Blue Party.    -   5. The magnitude of the difference between items 1 and 3 above        (the “Red-Blue Control Differential”).    -   6. The magnitude of the difference between items 2 and 4 above        (the “Lowest Voter Margin Differential”).    -   7. The number of tied districts (with a voter margin of 0).

Identifying the Winner(s)

The winner is identified by process of elimination. First, every playerwhose district plan violates one of the requirements A-B (seedescription of Example 19) is eliminated. Second, among the remainingplayers, every player whose district plan does not tie for having thelowest score for item 5 above is eliminated. Next, among the remainingplayers, every player whose district plan does not tie for having thelowest score for item 6 above is eliminated. Finally, among theremaining players, every player whose district plan does not tie forhaving the highest score for item 7 above is eliminated. Any player whois not eliminated wins the game. If all players' district plans violateone of the requirements A-B (see description of Example 19), all playerslose.

Example 21

Strategy games of Example 21 combine the alternating, turn-based playfor more than two players from Example 11 with the triangular sectors ofExample 15. The taxonomic code for these games isA/G/T1/P/3/9/U/2-3/EXR. They are analog, multi-player games with 54triangular sectors and a political focus in which three types ofelements—three political parties—are present and nine districts areformed. The games proceed according to alternating, turn-based play;there are 2-3 players; and moves in categories “E,” “X,” and “R” areallowed.

Game Summary

FIGS. 50-53 illustrate elements of one example of this kind of game, astrategy game 1200. In this game, 2-3 players representing opposingpolitical parties—Red, Green, and Blue—vie for political control of ahexagonal region 1202 by competitively creating nine political districtsout of 54 triangular sectors 1204 in turn-based fashion (FIG. 50). Thesectors 1204 may be pre-established on the region 1202, or they may beestablished during game set-up by placing sector tiles 1206 on sectorplaceholders (not shown) on the region 1202.

FIG. 51 shows two examples of sector tiles 1208 and 1210 for this game.The first sector tile 1208 has a first set of elements 1212, and thesecond sector tile 1214 has a second set of elements 1214. As shown inFIG. 51, each sector tile 1208 and 110 contains three identicalrepresentations of the set of elements in that tile. Alternatively, eachset of elements may be represented once, as shown on sector tiles 1206,or any suitable number of times in any arrangement suitable to be viewedby the players. Each set of elements includes one symbol for eachelement type (e.g., Red, Green, and Blue Parties) and one number showingthe quantity (e.g., how many people in the sector support the Red,Green, and Blue Parties respectively). There are 6 people in eachsector. The distribution of element sets for one example of a set ofsector tiles is provided below, with the number of [Blue, Red, Green]Party supporters in a sector indicated in square brackets and the numberof sector tiles of that type shown in parentheses:

[6, 0, 0] (2) [5, 0, 1] (2) [0, 1, 5] (2) [2, 0, 4] (2) [3, 0, 3] (2)[1, 1, 4] (2) [2, 1, 3] (2) [0, 6, 0] (2) [1, 5, 0] (2) [4, 2, 0] (2)[0, 4, 2] (2) [0, 3, 3] (2) [3, 2, 1] (2) [1, 3, 2] (2) [0, 0, 6] (2)[1, 0, 5] (2) [4, 0, 2] (2) [0, 2, 4] (2) [4, 1, 1] (2) [3, 1, 2] (2)[1, 2, 3] (2) [5, 1, 0] (2) [0, 5, 1] (2) [2, 4, 0] (2) [3, 3, 0] (2)[1, 4, 1] (2) [2, 3, 1] (2)

The mechanics of this game are slightly different than in previousexamples. In this game, the same types of moves—1, 1A, 2, 2A, 3, and3A—are allowed during play, but there are additional restrictionsregarding their timing. In particular, all moves of type 1, 1A, 2, and2A must be completed during phase 1 of this game—i.e. the board must becompletely full—before the first move of type 3 or 3A is allowed to bemade in phase 2. The district size threshold in moves of type 2 alsodiffers from earlier examples.

FIG. 50 shows a possible initial board position of strategy game 1200.

FIG. 52 shows a scoreboard 1216 that may be used in strategy game 1200.The scoring method in this game is nearly identical to that in Example11. Each row 1218 of the scoreboard is used for tracking the politicalstatus of a different district.

Play may include the following four phases. Phases 2-3 are optional.

-   -   1. Build political districts using moves of type 1, 1A, 2, and        2A    -   2. Rebalance district sizes using moves of type 3A    -   3. Run an election (optional)    -   4. Identify the winner

Phase 1: Build Political Districts Using Moves of Type 1, 1A, 2, and 2A

This phase proceeds almost exactly like phase 1 in Example 15. The maindifferences are that in this game (a) turns alternate among up to threeplayers instead of two players and (b) only moves of type 1, 1A, 2, and2A are available.

All moves made in phase 1 must be of type 1, 1A, 2, or 2A below.

Play during the first phase is divided into four stages. During stage 1,only moves of type 1 and 2 are allowed. Play enters stage 2 if (i) nomore moves of type 1 exist and (ii) fewer than nine districts have beenestablished. During stage 2, only moves of type 1A and 2 are allowed.Play enters stage 3 immediately after the 9^(th) district isestablished. During stage 3, only moves of type 2 are allowed. Playenters stage 4 if no more moves of type 2 exist. During stage 4, onlymoves of type 2A are allowed. This phase of the game ends when everysector has been assigned to a political district.

The four types of legal moves during this phase of the game are asfollows. Note that the “5-sector district size restriction” and “no tilecapturing restriction” in moves of type 2 is slightly different than inExample 15 and other previous examples.

-   -   1 Establish a new district (by placing its home base marker) on        a sector with the restriction that (i) the sector is at least        three steps away from each previously placed home base marker        and (ii) there is space to grow this district to a size of six        connected sectors.    -   1A Establish a new district (by placing its home base marker) on        a sector that is within the largest connected open space on the        board. An open space is an area where no sector has been        assigned to a district. Among the sectors satisfying the above        criterion, select a sector that ties for being the most steps        away from a previously placed home base marker.    -   2 Expand a district (by placing one of its expansion markers on        a sector) so that (i) no sectors are captured, (ii) no district        is trapped, (iii) the district's new size is no greater than 5        sectors, and (iv) the district remains connected.    -   2A Expand the smallest expandable district (by placing one of        its expansion markers on a sector) with the restriction that (i)        no sectors are captured and (ii) the district remains connected.

Phase 2: Rebalance District Sizes Using Moves of Type 3A

This optional phase of the game is motivated by the need to keep thepopulations of real-world political districts nearly equal. In thisphase of the game, players take turns modifying the sector-to-districtassignments in order to better equalize the district sizes (which are aproxy for the district populations).

Players take turns beginning with the player to the left of the playerwho took the final turn during phase 1. During a player's turn, he/shemakes a move by changing the district to which one sector is assigned.This is done by removing the (home base or expansion) marker thatoccupies one sector tile and replacing it with an expansion marker of adifferent color. The net result is that one district loses a sector andone district gains a sector. The other seven districts remain unchanged.The player then updates the scoreboard to reflect the move that has beenmade.

Every move made during this phase of the game must be of type 3A:

-   -   3A Reassign a community from one district (say District X) to        another (say District Y). This move must meet four        requirements. (j) District Y must exist prior to this move. (k)        District Y must not be expandable prior to this move. (1)        District X must be at least 2 sectors larger than District Y        prior to this move. (m) Districts X and Y must each remain        connected.

This phase of the game concludes when no more moves of type 3A exist.

Phase 3: Run an Election

Phase 3 in this game is optional and is very similar to phase 2 inExample 11. During this phase of the game, the political status of eachdistrict is converted into a numerical likelihood of each party winningthe district, and an election is simulated by rolling dice.

Phase 4: Identify the Winner(s)

In the game's final phase, the overall winner is identified. If phase 3was played, the winner is the player whose party wins the most districtelections. If more than one party ties for winning the most districtelections, these parties together win the game and the result is a tie.

If phase 3 was not played, the winner is determined by identifying theparty that controls each district, i.e. the party with the most votersin each district. Each party receives 6 points for each district that itsolely controls; 3 points for each district that it jointly controlswith one other party; and 2 points for each district that it jointlycontrols with two other parties. The winner is the player whose partyhas more points than any other player's party. If parties represented bytwo or more players tie for having the most points, those playersjointly win.

Symmetric Games

A symmetric version of this game may be played if there is a desire toeliminate bias in the initial sector arrangement. A symmetric game hasthree additional rules compared to a regular game. Rule 1 guarantees aninitial sector arrangement that is symmetric, whereas rules 2 and 3minimize the possibility of a symmetric position during play. The threerules are as follows.

-   -   1. During the game setup, the sector tile arrangement must be        rotationally symmetric. In a rotationally symmetric sector        arrangement the number of (Blue, Red, Green) Party supporters in        every sector is the same as the number of (Red, Green, Blue)        Party supporters in the sector that is a 120 degree clockwise        rotation from it. Such an arrangement guarantees that the        initial map is unbiased, favoring no party.        -   FIG. 53 shows a rotationally symmetric sector arrangement.            Bold lines distinguish three diamond-shaped portions of the            board: the upper-right diamond 1220, lower-right diamond            1222, and the left diamond 1224. Note that the number of            (Blue, Red, Green) Party supporters in every sector is the            same as the number of (Red, Green, Blue) Party supporters in            the sector that is a 120 degree clockwise rotation from it.    -   2. The second, third, and fourth moves made during phase 1 (i.e.        the 1^(st) move made by the player who goes second, the 1^(st)        move made by the player who goes third, and the 2^(nd) move made        by the player who goes first) may not involve the placement of a        marker on a sector that is either a 120 degree clockwise or 120        degree counterclockwise rotation from a sector on which a marker        has been placed.    -   3. During phase 1, no move may expand an existing district so        that it has size six and it coincides with the small hexagon at        the center of the region.

Example 22

Strategy games of this example encompass several games that are playedwith more sectors than Example 21 but are otherwise very similar toExample 21.

The taxonomic codes for three possible games included in this exampleare listed below. All games are analog, multi-player games withtriangular sectors and a political focus in which three types ofelements—namely three political parties—are present, the game proceedsaccording to alternating turn-based play, there are 2-3 players, andmoves in categories “E,” “X,” and “R” are allowed.

-   -   A/G/T2/P/3/12/U/2-3/EXR    -   A/G/T3/P/3/15/U/2-3/EXR    -   A/G/T4/P/3/27/U/2-3/EXR

The three games above get progressively larger with 96, 150, and 216sectors respectively.

Example 23

Strategy games of Example 23 combine the simultaneous independent playof Example 19 with the three-party environment in Example 21. Thesegames have taxonomic code A/G/T1/P/3/9/I. They are analog, multi-playergames with 54 triangular sectors and a political focus in which threetypes of elements—three political parties—are present and nine districtsare formed. The game paradigm is simultaneous independent play, so anynumber of players may participate.

Components

In one example, the game components are similar to the components usedin Example 21. A digital or mechanical timer is needed to play thisgame. In addition, each player should have a copy of the same game setwhich contains (a) 54 sectors having the same markings as in Example 21,(b) dozens of markers, (c) a scoreboard, and (d) a game board.

Playing the Game

Play may include the following four phases. Phases 2-4 are optional.

-   -   1. Build political districts that maximize the Red Party's        advantage (time limit 10 min)    -   2. Build political districts that maximize the Green Party's        advantage (time limit 10 min)    -   3. Build political districts that maximize the Blue Party's        advantage (time limit 10 min)    -   4. Build political districts that equalize the advantage of all        parties (time limit 10 min)

Each phase 1-3 proceeds like a phase in Example 19. Phase 4 proceeds asin Example 20.

At the end of each phase of the game, each player tracks his/her scorewith respect to the goal at hand by placing markers on the appropriatesquares on his/her game board.

Identifying the Winner(s)

The winner is the player who does the overall best job of achieving thegoals that were pursued during the different phases of the game.

Example 24

Strategy games of Example 24 are played with hexagonal sectors. Someexamples of these games have taxonomic code A/G/H1/P/2/9/U/2/EXR. Theyare analog, multi-player games with 37 hexagonal sectors and a politicalfocus in which two types of elements—two political parties—are presentand nine districts are formed. Taxonomic codes for five additional gamesincluded in this example are listed below. All five games are analog,multi-player games with hexagonal sectors and a political focus in whichtwo types of elements—two political parties—are present, the gameproceeds according to alternating turn-based play, and there are 2players.

-   -   A/G/H2/P/2/15/U/2/EXR    -   A/G/H3/P/2/15/U/2/EXR    -   A/G/H4/P/2/21/U/2/EXR    -   A/G/H5/P/2/21/U/2/EXR    -   A/G/H6/P/2/27/U/2/EXR

The five games above get progressively larger with 61, 91, 127, 169, and217 sectors respectively. The shape of the region in each game isessentially a regular hexagon.

The strategy games of this example proceed according to alternating,turn-based play; there are two players; and moves in categories “E,”“X,” and “R” are allowed.

FIG. 54 shows a region 1302 having 37 sector placeholders 1304. Sectortiles 1306, two examples of which shown in FIG. 55, may be placed on theregion, one on each sector placeholder 1304, to form the sectors. Eachsector has a set of elements 1308. During play, nine districts (whoseaverage size is roughly 4) are formed from the sectors. It should beunderstood that regions of these games can be any size, and contain anynumber of sectors. The number of districts to be formed may varydepending upon the number of sectors in the region.

Rules that can be used for strategy games of this type are generallysimilar or identical to those of previous examples.

Example 25

Strategy games of Example 25 have taxonomic code A/G/H1/P/2/9/I. Theyare analog, multi-player game with 37 hexagonal sectors and a politicalfocus in which two types of elements—two political parties—are presentand nine districts are formed. These games proceed according tosimultaneous independent play, so any number of players may participate.

Rules that can be used for strategy games of this type are generallysimilar or identical to those of previous examples with two parties andsimultaneous independent play.

Example 26

Strategy games of Example 26 have taxonomic code A/G/H1/P/6/9/U/2-6/EXR.They are analog, multi-player games with 37 hexagonal sectors and apolitical focus in which six types of elements—six political parties—arepresent and nine districts are formed. These games proceed according toalternating, turn-based play; there are 2-6 players; and moves incategories “E,” “X,” and “R” are allowed.

Rules that can be used for strategy games of this type are generallysimilar or identical to those in Examples 11 and 21.

Example 27

Strategy games of Example 27 have taxonomic code A/G/H1/P/6/9/I. Theyare analog, multi-player games with 37 hexagonal sectors and a politicalfocus in which six types of elements—six political parties—are presentand nine districts are formed. These games proceed according tosimultaneous independent play, and a variety of pre-defined goals can bepursued in each phase of these games.

Rules that can be used for strategy games of this type are generallysimilar or identical to those in Examples 13-14 and 23.

Example 28

Strategy games of Example 28 have a real-world focus in which U.S.congressional districts are formed in a real U.S. state, namelyWisconsin. These games have taxonomic code A/G/C82/P/3/8/U/2/EXR. Theyare analog, multi-player games with 82 complex sectors and a politicalfocus in which three types of elements—population, Red Party supporters,and Blue Party supporters—are considered and eight districts are formed.These games proceed according to alternating, turn-based play; there are2 players; and moves in categories “E,” “X,” and “R” are allowed.

Game Summary

FIGS. 56-70 illustrate aspects of a strategy game 1400. In this game,two players representing opposing political parties (Red and Blue) viefor political control of the region 1402 (i.e., the state of Wisconsin)by competitively creating eight U.S. congressional districts out of 82sectors 1404 that largely coincide with the counties currently existingin Wisconsin. Wisconsin has a two-party political system in which oneperson is elected to the U.S. House of Representatives to represent eachcongressional district. As shown in FIG. 56, at the outset, thedistricts are formless and the players know the shape, location,population, and political composition of each sector (i.e. which partyits citizens favor and by how much) based on the set of elements 1408depicted in each sector 1404. During the first phase of the game,players gradually build political districts by assigning sectors topolitical districts one sector at a time. They may also reassign sectorsfrom more populated districts to less populated districts in order tobetter equalize the district populations. During the (optional) secondphase of the game, the political margin in each district is convertedinto a probability of each party winning the district, and an electionis simulated by rolling dice (such as dice 1622 of FIG. 44). The winnerof the game is the player whose party controls more districts thanhis/her opponent. If both players control an equal number of districts,the result is a tie.

Components

The components of the game are listed below:

-   -   Region 1402 (Wisconsin map) having 82 sectors 1404, each sector        having a set of elements 1408    -   8 marker subsets (one for each of the eight districts—Brown,        Red, Orange, Yellow, Green, Blue, Purple, and Gray), each        containing a plurality of markers 1418, which may include, for        example, 8 home base markers 1422 (one for each district) and        280 expansion markers 1424 (35 for each of the eight districts)    -   64 scoring tokens (eight for each of the eight districts above)    -   2 ten-sided dice (one black, one white) each showing values 0-9    -   Two-part scoreboard having a first part 1410 (FIG. 57) and a        second part 1412 (FIG. 58).

Setup

The players decide (yes or no) if phase 2 of the game will be played.The players then decide who plays Red and who plays Blue, and who willtake the first turn.

Sectors

As can be seen in FIG. 56, each sector on the map contains two numbers.In accordance with the key in Table 1, the first number 1414 (in arectangle) denotes the sector's voting population (in thousands). Thesecond number 1416 indicates the sector's voter margin (i.e. votingtendency, political margin). In the second number 1416, a black numberin a white circle indicates that the sector tends to vote for the RedParty; a white number in a black circle indicates that the sector tendsto vote for the Blue Party. The number itself, in this example, is thevoting margin (in thousands of votes) by which the sector supported oneparty over the other in the 2016 U.S. presidential election. Forexample, a black 4 in a white circle indicates that the sector supportedthe Red Party by a margin of 4000 votes in the 2016 election. A white 0in a black circle indicates that voters in the sector were evenlydivided—after rounding off to the nearest thousand voters—among the twoparties in the 2016 U.S. presidential election.

In this game, the 82 sectors 1404 are used as building blocks to formeight non-overlapping political districts which together exhaust theland area of the state. The eight districts are identified by color:Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray. Initially,the political districts are formless and no sector belongs to anydistrict. During the course of the game, players use markers 1418 togradually assign these 82 sectors to political districts. Each sectoreventually belongs to exactly one political district.

A close inspection of the sectors 1404 as shown in FIG. 56 will revealthat the sum of the black numbers in white circles exceeds the sum ofthe white numbers in black circles by 23. In other words, there are23,000 more Red Party supporters than Blue Party supporters in theregion 1402. This value agrees with the results of the 2016 U.S.presidential election. Based on this information, we say that theoverall political margin in the state is “+23 for the Red Party” or “+23Red.” Although the Red Party has a slight advantage statewide, it ishighly unlikely that the Red Party will be able to maintain an advantagewithin every district after the region 1402 is divided into eightdistricts. Also note that the sum of the numbers in the black rectanglesis 2793. In other words, the state's voter population is 2,793,000.Dividing this value by 8, we see that, at the game's end, the averagedistrict's voter population will be 349.125, which is most closelyrepresented by the value 349. However, it is unlikely that any districtswill have a population of exactly 349 at the game's end; most or alldistricts will have a population strictly greater than or less than 349.

Scoreboard

Scoring tokens are used to display the (voter) population and politicalmargin of every district on scoreboard (FIGS. 57-58) at all times. Atthe start of the game, these scoring tokens may be placed to show thatthe population of each district is zero and that no party has anadvantage in any district.

The scoreboard should be updated after every player takes a turn. Forexample, consider a moment in the game when exactly three sectors withpopulations 40, 71, and 48 and voting tendencies “+7 Red,” “+8 Blue,”and “+10 Red” respectively have been assigned to the Green District. Inthis case, the Green District's population—159—should be indicated bythree green scoring tokens placed at positions “Population x100=1,”“Population x10=5,” and “Population x1=9” in the Green District'sportion of the scoreboard. The Green District's current politicalmarging—“+9 Red”—equals the difference between the sum of the numbers inthe sectors that support the Red Party (e.g. 17) and the sum of thenumbers in the sectors that support the Blue Party (e.g. 8) that belongto the district. The political margin favors the Red Party if there aremore Red Party than Blue Party supporters in the district; it favors theBlue Party if the opposite is true. A district's political margin isindicated by four scoring tokens that show (1) which party has themajority of voters in the district, (2) the hundreds digit of thepolitical margin, (3) the tens digit of the political margin, and (4)the ones digit of the political margin. In the case above, four greenscoring tokens should be placed at the positions “Current Leader=Red,”“Political Margin x100=0,” “Political Margin x10=0,” and “PoliticalMargin x1=9” in the Green District's portion of the scoreboard. If asector with population 72 and voting tendency “+36 Blue” is added tothis district, the district's new population is 231 and its newpolitical margin is “+27 Blue,” so the seven green scoring tokens shouldimmediately be moved to positions “Population x100=2,” “Populationx10=3,” “Population x1=1,” “Current Leader=Blue,” “Political Marginx100=0,” “Political Margin x10=2,” and “Political Margin x1=7.”

Playing the Game

Play may include the following three phases (the second phase isoptional):

-   -   1. Build political districts    -   2. Run an election (optional)    -   3. Identify the winner

Phase 1: Build Political Districts Summary

This is the main phase of the game. During this phase, players taketurns assigning sectors 1404 to political districts, one sector at atime, until every sector 1404 belongs to a political district. Theassignment of a sector to a political district is accomplished byplacing a home base marker or expansion marker on a sector. Players mayalso reassign sectors from more populated districts to less populateddistricts in order to better equalize the district populations. This isdone by changing the color of the marker that occupies a sector. At theend of this phase, there will be eight non-overlapping politicaldistricts—Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray—thattogether cover the state. Also, each district will be connected.

Each district evolves in the same general way. Initially, it isformless. At some point, the district is established when its home basemarker is placed on a vacant sector. (A vacant sector is a sector withno marker on it.) The district is then expanded whenever one of itsexpansion markers is placed on a vacant sector that is adjacent to asector that already belongs to the district. Later, the district may beadjusted so its population is more similar to neighboring districts.

The overall process of building the political districts is relativelyunrestricted. In general, any player may contribute to building anydistrict during any of his/her turns. There is no requirement for thesequence in which, or locations where, districts are constructed. Oncebegun, the construction of a given district may be temporarily haltedwhile players take turns establishing, expanding, and/or adjusting otherdistricts. There is no district population requirement. However, therules encourage the creation of districts whose population is close tothe average value of 349.

Details

Players take turns beginning with the starting player. During a player'sturn, he/she (A) makes one move and then (B) records the move on thescoreboard. Forfeiting a turn is not allowed.

The rules of this game provide that all moves made during this phase ofthe game must be of type 1, 1A, 2, 2A, 3, or 3A below. Moves of type 1and 1A establish a new district. Moves of type 2 and 2A expand anexisting district. Moves of type 3 and 3A adjust two adjacent districtsby transferring a sector from one district to an adjacent district. “A”stands for “alternate move.”

Phase 1 of the game is divided into three stages. During stage 1, onlymoves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) nomore moves of type 1 exist and (ii) fewer than eight districts have beenestablished. During stage 2, only moves of type 1A, 2, and 3 areallowed. Play enters stage 3 immediately after the 8^(th) district isestablished. During stage 3, the next move must be of type 2 or 3A if atleast one move of type 2 exists. Otherwise, the next move must be oftype 2A or 3A. This phase of the game ends when no more legal movesexist.

The six types of legal moves are as follows. Explanations for theasterisked terms are provided at the end of these descriptions.

-   -   1 Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (i) The        sector must be at least three steps away from all previously        placed home base markers. (ii) There must be space to grow this        district into a connected* district with a population of at        least 349. (In FIG. 59, sectors 11 and 34 are three steps away        from each other. Sectors 25 and 44 are two steps away from each        other. Also, sectors 57 and 72 are two steps away from each        other.)    -   1A Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (i) The        sector must be in the most populated open space on the board.        (An open space is a connected* group of vacant sectors.) (ii)        Among the sectors satisfying the criterion above, the sector        must be the farthest (in number of steps) from a previously        placed home base marker. Ties may be broken arbitrarily.    -   2 Expand a district by placing one of its expansion markers on a        vacant sector. This move must meet four requirements. (i) The        district must remain connected.* (ii) The district's initial        population before the move must be 348 or less. (iii) The        district's new population—including the new sector and any        sectors that are captured**—must be 398 or less. (iv) No        district may be trapped.***    -   2A Expand the least populous expandable district by placing one        of its expansion markers on a vacant sector. The district must        remain connected* and no sectors may be captured.** (A district        is expandable if there is at least one vacant sector adjacent to        it.)    -   3 Reassign a sector from one district (e.g. District X) to        another (e.g. District Y) by removing the District X marker from        a sector and replacing it with a District Y expansion marker.        This move must meet four requirements. (i) Districts X and Y        must remain connected.* (ii) The populations of Districts X and        Y must become strictly more balanced. In other words, before        this move is made, the population of X must exceed the        population of Y by more than the population of the reassigned        sector. (iii) District Y must exist and be confined, i.e. it        must not be expandable, prior to this move. In other words,        there must be no vacant sectors adjacent to District Y prior to        this move. (iv) The District X marker that is removed must be an        expansion marker; it may not be a home base marker.    -   3A Same as move type 3 but without requirement (iv) for move        type 3.

-   * See subsection entitled “Connectedness” below

-   * See subsection entitled “Captured sectors” below

-   ** See subsection entitled “Trapped districts” below

Connectedness

Two sectors are adjacent—and connected—if and only if they share acommon edge. For example, in the map shown in FIG. 59 (in which each ofthe 82 sectors 1404 has been given a reference number 1420 for ease ofreference), sectors 1 and 2 are adjacent; sectors 35 and 45 areadjacent; and sectors 69 and 74 are adjacent. However, sectors 1 and 8are not adjacent; sectors 44 and 56 are not adjacent; and sectors 30 and41 are not adjacent. By analogy the territory consisting of sectors 25,33, 44, and 54 is connected, and the territory consisting of sectors 57,64, 68, and 72 is connected. The territory consisting of sectors 59 and68-70 is not connected.

In this game, every political district must be connected at all times.That is, at all times and for any two sectors that belong to a givendistrict (say District X), there must be a path within District X (i.e.a sequence of adjacent sectors that all belong to District X) connectingthose two sectors.

Captured Sectors

A set of connected, vacant sectors is captured if it is (i) surroundedby the edge of the board on one side and a single district on the otherside or (ii) entirely surrounded by a single district. In FIG. 59,sectors 60 and 76 are captured by the Green District (consisting ofsectors 54, 61-62, and 77); sector 18 is captured by the Blue District(consisting of sectors 17, 19, and 24); sector 41 is captured by theGray District (consisting of sector 40); sector 29 is captured by theYellow District (consisting of sectors 23, 28, and 30); and sectors73-75 are captured by the Red District (consisting of sectors 69-70, 72,and 81).

A move of type 2 which captures one or more sectors is allowed if thedistrict's initial population before the move is 348 or less and thedistrict's new population—including the sector where the marker isplaced and any sectors that are captured—is 398 or less. All other movesthat capture sectors are forbidden. For example, in FIG. 59, it ispermissible to add sector 54 to a district consisting of sectors 61-62and 77. In this case, the district immediately grows to include sectors54, 60-62, and 76-77 after the move is made, and the district's newpopulation is 68 which is well below the 398 threshold (FIG. 56 showsthe sector populations). However, it is not permissible to add sector 72to a district consisting of sectors 69-70 and 81 because the expandeddistrict—which after capturing three sectors would consist of sectors69-70, 72-75, and 81—would have a population of 569 which is well abovethe 398 threshold.

Any sectors that are captured during a legal move of type 2 areimmediately assigned to the district that has captured them. Expansionmarkers are immediately placed on these sectors.

Trapped Districts

A district is trapped if (i) it (and the open spaces beside it) issurrounded either by the edge of the board on one side and a singledistrict on the other side or by a single district on all sides and (ii)its population plus the populations of all open spaces beside it is 348or less. In FIG. 59, the Orange District (consisting of sector 11) istrapped by the Brown District (consisting of sectors 8, 10, 12, 20, 22,26-27, and 34) because the combined population of sectors 11 and 21(=16) is 348 or less (FIG. 56 shows the sector populations).

A move of type 2 which traps a district is forbidden. For example, ifthe Brown District consists of sectors 10, 12, 20, 22, 26-27, and 34 andthe Orange District consists of sector 11, then an expansion of theBrown District to sector 8 is forbidden.

Phase 2: Run an Election

In this phase, the political margin in each district is converted into anumerical likelihood of each party winning the district, and an electionin each district is simulated by rolling the two 10-sided dice.

Each district is considered one at a time. First, using the table below,the political margin for the party with more voters in the district isconverted into a numerical likelihood of that party winning an electionin the district. For example, a “+4 Blue” political margin in the YellowDistrict converts to a 66% chance for the Blue Party to win an electionin the Yellow District.

Political Winning Margin Likelihood 0 50% +1 54% +2 58% +3 62% +4 66% +570% +6 73% +7 76% +8 79% +9 82% +10 85% +11 87% +12 89% +13 91% +14 93%+15 95% +16 96% +17 97% +18 98% +19 99% +20 or more 100% 

Next, a random number from 1-100 is produced by rolling the two 10-sideddice.

The random number is then compared to the winning percentage (e.g. 66for the above case). If the random number is less than or equal to thewinning percentage, the party with more voters in the district wins thedistrict election. If the random number is greater than the winningpercentage, the party with fewer voters in the district wins thedistrict election. In the above example, the Blue Party wins the YellowDistrict election if the random number is from 1-66, and the Red Partywins the Yellow District election if the random number is from 67-100.If both parties have a 50% chance of winning the election, the BlueParty wins if the random number is 1-50 and the Red Party wins if therandom number is 51-100. After the winner of an election is identified,the unused scoring token that matches the district color is placed onthe “Winner=Blue” or “Winner=Red” square in the appropriate district'sportion of the scoreboard.

The above procedure is repeated for each political district.

Phase 3: Identify the Winner

In the game's final phase, the overall winner is identified.

If phase 2 is played, the winner is the player whose party wins five ormore district elections. If each party wins four district elections, theresult is a tie.

If phase 2 is not played, the scoreboard is used to identify the partythat controls each district, i.e. the party with more voters in eachdistrict. The winner is the player whose party controls more districtsthan his/her opponent. If the two players control an equal number ofdistricts, the result is a tie.

Example of Play

An example of play is now provided, with reference to FIGS. 60-70, toillustrate the rules of the game. The initial position is shown in FIG.56. During play, the population and voting tendency of each sector arevisible to both players. However, in FIGS. 60-69, the reference number1420 of each sector 1404 is shown instead of the voting tendency forease of discussion.

During stage 1, only moves of type 1, 2, and 3 are allowed. After 40moves have been made—six of type 1 and 34 of type 2—assume the boardposition, with home base markers 1422 and expansion markers 1424, is asshown in FIG. 60. Note that six districts—Brown, Red, Orange, Yellow,Blue, and Purple—have been established. Two districts—Green andGray—have not (yet) been established.

There are 43 possibilities for the next move which must be of type 1, 2,or 3. No legal moves of type 3 exist because no district is confined.Regarding moves of type 1, it is possible to establish a new district insector 73 or 74. Establishing a new district in another sector is notallowed because either (i) the sector is less than three steps away froma previously placed home base marker or (ii) there is not enough spaceto grow the new district into a connected district with a population ofat least 349. Regarding moves of type 2, it is possible to (a) expandthe Brown District to sector 5, 13, 22, 21, 19, 56, 50, 46, 28, 29, 30,or 16; (b) expand the Red District to sector 28, 46, 51, 57, 53, 40, 38,or 37; (c) expand the Orange District to sector 19, 24, 31, 32, 42, 43,54, 60, 76, 65, 63, or 56; (d) expand the Yellow District to sector 57,67, 70, 72, or 59; or (e) expand the Blue District to sector 4, 9, 21,or 12. The Purple District may not be expanded during the next movebecause its population—352—is already at least 349.

Several of the above moves of type 2 capture one or more sectorsincluding the expansion of the (a) Brown District to sector 30 (whichcaptures sector 16); (b) Red District to sector 40 (which capturessector 41); (c) Red District to sector 38 (which captures sectors 40 and41); and (d) Orange District to sector 24, 31, 32, 42, 54, or 60. Notethat an expansion of the Brown District to sector 9 is not allowedbecause the Blue District would be trapped (by the Brown District).Also, an expansion of the Yellow District to sector 53, 73, or 74 is notallowed because one or more sectors would be captured and added to theYellow District, putting its population over the limit of 398.

The next move played is the establishment of the Green District insector 74. This results in the game position shown in FIG. 61. In thisposition, (i) there is no way to make a move of type 1 that satisfiesits criteria and (ii) fewer than eight districts have been established.Thus, stage 2 of play begins.

There are 52 possibilities for the next move which must be of type 1A,2, or 3. No legal moves of type 3 exist because no district is confined.The feasible moves of type 2 include the 41 moves of type 2 mentionedabove and (f) expanding the Green District to sector 70, 75, or 73.

A move of type 1A requires that a new district be established in asector that is within the most populous connected open space on theboard. Among the sectors satisfying this criterion, a sector that tiesfor being the most steps away from a previously placed home base markermust be selected. In the current board position, the most populous openspace has population 511 and consists of sectors 46, 50-51, 56-57,63-65, 67, 70, and 75. Among these sectors, eight tie for being twosteps away from a previously placed home base marker—46, 50-51, 56,63-65, and 67—and none is three or more steps away from all previouslyplaced home base markers. Thus, there are eight possible moves of type1A: establish the Gray District in sector 46, 50-51, 56, 63-65, or 67.

The next five moves in this example game are as follows (move type inparentheses):

-   -   1. (2) Expansion of Red District to sector 40 (one sector        captured)    -   2. (2) Expansion of Blue District to sector 12    -   3. (2) Expansion of Brown District to sector 30 (one sector        captured)    -   4. (2) Expansion of Orange District to sector 24 (seven sectors        captured)    -   5. (1A) Establishment of Gray District in sector 56

The establishment of the 8^(th) district during move #5 above ushers instage 3 of play. During stage 3, the next move must be of type 2 or 3Aif at least one type 2 move exists. Otherwise the next move must be oftype 2A or 3A. The new board position is shown in FIG. 62.

There are 40 possibilities for the next move. Regarding moves of type 2,it is possible to (a) expand the Brown District to sector 5, 13, 22, 21,19, 50, 46, 28, 29, or 38 (but not 9); (b) expand the Red District tosector 28, 46, 51, 57, 53, 38, or 37; (c) expand the Orange District tosector 18, 65, 63, or 19; (d) expand the Yellow District to sector 57,67, 70, 72, or 59 (but not 53 or 73); (e) expand the Green District tosector 70, 75, or 73; (f) expand the Blue District to sector 4, 9, 21,22, 13, or 5; or (g) expand the Gray District to sector 50, 63, 64, 57,or 51. The Purple District may not be expanded during the next movebecause its population—352—is already at least 349. There are no legalmoves of type 3A because no district is confined.

After the next 23 moves in the game, the exemplary board position isshown in FIG. 63. The preceding 23 moves were all of type 2, and thesemoves involved the expansion of seven districts: Brown, Red, Orange,Yellow, Green, Blue, and Gray.

The table below shows the current population of each district. Note thatno feasible move of type 2 exists. This is because all districts alreadyhave a population of at least 349 and/or are confined. In particular,the only two districts with a population of 348 or less—Blue andGray—are confined. Thus, the next move must be of type 2A or 3A.

District Population Brown 366 Red 359 Orange 360 Yellow 355 Green 352Blue 243 Purple 352 Gray 141

A move of type 3A is possible—i.e. a sector may be reassigned fromDistrict X to District Y—if and only if (i) Districts X and Y remainconnected after the reassignment, (ii) before the reassignment thepopulation of X exceeds the population of Y by more than the populationof the reassigned sector, and (iii) District Y is confined before thereassignment. Note that accurate district population information (asshown in the table above) is needed in order to make a correctassessment regarding item (ii) above.

To search for moves of type 3A, we consider each district one at a timeand ask if that district can “steal” a sector from another district.

-   -   The Brown District may not steal a sector from another district        for two reasons: (A) it is the most populous district and (B) it        is not confined.    -   The Red District is not confined, so it may not steal a sector        from another district.    -   The Orange District is confined and its population is 6 less        than the Brown District, so the Orange District could        theoretically steal a sector with population 5 or less from the        Brown District. However, the Brown District has no such sector        that lies along the Orange-Brown border, so there is no legal        way for the Orange District to steal a sector from another        district.    -   The Yellow District is not confined, so it may not steal a        sector from another district.    -   The Green District is not confined, so it may not steal a sector        from another district.    -   There are five legal moves of type 3A in which the Blue        District—whose population is much less than the other districts        except the Gray District—steals a sector from a neighboring        district. These include (a) reassigning sector 10, 14, or 23        from the Brown District to the Blue District and (b) reassigning        sector 24 or 25 from the Orange District to the Blue District.        Note that reassigning sector 20, 26, or 27 from the Brown        District to the Blue District destroys the connectedness of the        Brown district, so these moves are not allowed.    -   The Purple District's population is slightly less than that of        the Brown, Red, Orange, and Yellow Districts, so the Purple        District could theoretically steal a small sector from one of        these districts. However, the Purple District does not share a        border with the Brown, Red, or Yellow District. In addition, the        Orange District does not have a small sector that lies along the        Purple-Orange border. Thus, there is no legal way for the Purple        District to steal a sector from another district.    -   There are eight legal moves of type 3A in which the Gray        District—the least populous district—steals a sector from        another district. These include (a) reassigning sector 51 from        the Brown District to the Gray District; (b) reassigning sector        55, 62, or 65 from the Orange District to the Gray District; (c)        reassigning sector 66 from the Purple District to the Gray        District; (d) reassigning sector 64 from the Green District to        the Gray District; (e) reassigning sector 58 from the Yellow        District to the Gray District; and (f) reassigning sector 52        from the Red District to the Gray District. The following moves        are not allowed because they destroy the connectedness of a        district: (g) reassigning sector 45 or 50 from the Brown        District to the Gray District; (h) reassigning sector 67 from        the Green District to the Gray District; and (i) reassigning        sector 68 from the Yellow District to the Gray District.

To search for moves of type 2A, note that the Green District—with apopulation of 352—is the least populous expandable district. Only onemove of type 2A is available: (a) expand the Green District to sector73. Overall, a total of 14 legal moves exist.

The next five moves are as follows (move type in parentheses):

-   -   1. (2A) Expansion of Green District to sector 73    -   2. (3A) Reassignment of sector 23 from Brown District to Blue        District    -   3. (3A) Reassignment of sector 10 from Brown District to Blue        District    -   4. (3A) Reassignment of sector 24 from Orange District to Blue        District    -   5. (3A) Reassignment of sector 51 from Brown District to Gray        District

The board position is now as shown in FIG. 64. The current populationsof the districts are shown in the table below. Districts with apopulation of 348 or less are asterisked. Note that two districts with apopulation of 348 or less—Brown and Gray—are not confined. (The GrayDistrict—the least populated district—became unconfined during move #5above. The Brown District—which was never confined—becameunderpopulated—with a population of 348 or less—during move #3 above.)Thus, a move of type 2 is available. The next move must therefore be oftype 2 or 3A.

District Population Brown  325* Red 359 Orange 356 Yellow 355 Green 435Blue  279* Purple 352 Gray  150*

Two moves of type 2 are available: (a) expand the Gray District tosector 46 and (b) expand the Brown District to sector 37. Six moves oftype 3A are also available: (c) reassign sector 14, 29, or 20 from theBrown District to the Blue District; (d) reassign sector 25 or 31 fromthe Orange District to the Blue District; and (e) reassign sector 64from the Green District to the Purple District. Note that the Brown,Red, Yellow, Green, and Gray Districts are not confined, so they may notsteal a sector from another district. Also, the Purple District may notsteal sector 75 from the Green District because the population imbalancewould not be reduced. Overall, eight legal moves are available.

The next five moves are as follows (move type in parentheses):

-   -   1. (2) Expansion of Brown District to sector 37    -   2. (2) Expansion of Gray District to sector 46    -   3. (3A) Reassignment of sector 20 from Brown District to Blue        District    -   4. (3A) Reassignment of sector 31 from Orange District to Blue        District    -   5. (3A) Reassignment of sector 26 from Brown District to Blue        District

The new board position is shown in FIG. 65. The current districtpopulations are shown in the table below. Districts with a population of348 or less are asterisked. There are no feasible moves of type 2, sothe next move must be of type 2A or 3A.

District Population Brown 370 Red 359 Orange 349 Yellow 355 Green 435Blue  329* Purple 352 Gray  161*

There are 17 possibilities for the next move. To search for moves oftype 2A, note that the Yellow District—with a population of 355—is theleast populous expandable district. Only one move of type 2A isavailable: (a) expand the Yellow District to sector 72. Several moves oftype 3A are available: (b) reassign sector 14, 29, or 34 from the BrownDistrict to the Blue District; (c) reassign sector 32 or 33 from theOrange District to the Blue District; (d) reassign sector 64 from theGreen District to the Purple District; (e) reassign sector 35 or 50 fromthe Brown District to the Gray District; (f) reassign sector 36 or 52from the Red District to the Gray District; (g) reassign sector 55, 62,or 65 from the Orange District to the Gray District; (h) reassign sector58 from the Yellow District to the Gray District; (i) reassign sector 64from the Green District to the Gray District; and (j) reassign sector 66from the Purple District to the Gray District.

The next four moves are as follows (move type in parentheses):

-   -   1. (3A) Reassignment of sector 64 from Green District to Gray        District    -   2. (3A) Reassignment of sector 34 from Brown District to Blue        District    -   3. (3A) Reassignment of sector 62 from Orange District to Gray        District    -   4. (2A) Expansion of Yellow District to sector 72

The new board position is shown in FIG. 66. Note that there are novacant sectors. Thus, all future moves will be of type 3A. Playcontinues until no more such moves exist. The new district populationsare shown in the table below.

District Population Brown  334* Red 359 Orange  338* Yellow 438 Green363 Blue 365 Purple 352 Gray  244*

There are 21 possibilities for the next move: (a) reassign sector 5, 13,21, 22, or 23 from the Blue District to the Brown District; (b) reassignsector 58 or 59 from the Yellow District to the Red District; (c)reassign sector 19 or 31 from the Blue District to the Orange District;(d) reassign sector 68 or 69 from the Yellow District to the GreenDistrict; (e) reassign sector 50 from the Brown District to the GrayDistrict; (f) reassign sector 36 or 52 from the Red District to the GrayDistrict; (g) reassign sector 55, 61, or 65 from the Orange District tothe Gray District; (h) reassign sector 58 or 68 from the Yellow Districtto the Gray District; (i) reassign sector 67 from the Green District tothe Gray District; or (j) reassign sector 66 from the Purple District tothe Gray District.

The next two moves are as follows (move type in parentheses):

-   -   1. (3A) Reassignment of sector 22 from Blue District to Brown        District    -   2. (3A) Reassignment of sector 68 from Yellow District to Gray        District

The new board position is shown in FIG. 67. Note that the new districtpopulations—shown in the table below—are more balanced than before. Eachfuture move will make the district populations even more balanced until,at the end of the game, no further “rebalancing” is possible. At thatpoint, an “equilibrium” will be established, and the competition betweenthe two players—who alternate turns and are representing the Red andBlue Parties—will end.

District Population Brown  348* Red 359 Orange  338* Yellow 364 Green363 Blue 351 Purple 352 Gray  318*

There are five possibilities for the next move: (a) reassign sector 31from the Blue District to the Orange District; (b) reassign sector 50from the Brown District to the Gray District; (c) reassign sector 36from the Red District to the Gray District; (d) reassign sector 61 fromthe Orange District to the Gray District; or (e) reassign sector 67 fromthe Green District to the Gray District.

The next two moves are as follows (move type in parentheses):

-   -   1. (3A) Reassignment of sector 36 from Red District to Gray        District    -   2. (3A) Reassignment of sector 31 from Blue District to Orange        District

The new board position is shown in FIG. 68 and the new districtpopulations are in the table below.

District Population Brown 348* Red 334* Orange 345* Yellow 364  Green363  Blue 344* Purple 352  Gray 343*

There is only one possibility for the next move: reassign sector 29 fromthe Brown District to the Blue District. This move is compulsory for theplayer who takes the next turn.

The new board position is shown in FIG. 69 and the new districtpopulations are in the table below. No legal moves exist in this boardposition, so phase 1 of play concludes. The districts that have beenformed are now final. All districts are connected, but some—such asBrown, Yellow, Green, Blue, and Gray—have irregular shapes withtentacle-like protrusions. As expected, the district populations sum to2793, and the population of the average district is 349.125=2793/8.However, no district has a population equaling 349 or 350. Threedistricts—Yellow, Green, and Purple—have populations exceeding 350 andfive districts—Brown, Red, Orange, Blue, and Gray—have populations below349. The district populations are not perfectly smooth, but they arerelatively balanced. The population difference between the most populousdistrict—Yellow—and least populous district—Red—is only 30. Thissituation is not uncommon at the end of phase 1.

The final board position at the end of phase 1 is shown in FIG. 70. Themarkers played and each sector's population and voting tendency—whichare always displayed to both players during the game—are shown. Thepopulation is displayed in a black rectangle and the voting tendency isdisplayed in a circle.

At the end of phase 1, the scoreboard should read as shown in the boxbelow.

Political District Population Margin Brown 347 +61 Red  Red 334 +56 Red Orange 345 +31 Blue Yellow 364  +4 Blue Green 363 +69 Blue Blue 345 +64Red  Purple 352 +30 Blue Gray 343 +24 Blue Overall 2793 +23 Red 

If phase 2 is not played, the game ends, and the Blue Party wins by ascore of 5 districts to 3 districts.

If phase 2 is played, an election is simulated. In the final scoreboard(see above), the political margin of the (Brown, Red, Orange, Yellow,Green, Blue, Purple, Gray) District translates to a winning likelihoodof (100%, 100%, 100%, 66%, 100%, 100%, 100%, 100%) for the party thathas the majority of voters in the district. Note that the Red Partyautomatically wins three districts with 100% probability and the BlueParty automatically wins four districts.

Dice are then thrown to determine the election results for the districtin which there is not an automatic winner. The results are summarized inthe table below. Despite being at a disadvantage going into theelection, the Red Party “gets lucky” and wins the elections in fourdistricts. The result is a tie.

Political Winning Dice Election District Margin Likelihood Roll ResultBrown +61 Red  100% — Red Wins Red +56 Red  100% — Red Wins Orange +31Blue 100% — Blue Wins Yellow  +4 Blue  66% 67 Red Wins Green +69 Blue100% — Blue Wins Blue +64 Red  100% — Red Wins Purple +30 Blue 100% —Blue Wins Gray +24 Blue 100% — Blue Wins

Example 29

Strategy games of this Example 29 have a real-world focus in which U.S.congressional districts are formed in a real U.S. state, namelyMichigan. These games have taxonomic code A/G/C108/P/3/14/U/2/EXR. Thesegames are analog, multi-player games with complex sectors (which maymimic actual counties of the state) and a political focus in which threetypes of elements—population, Red Party supporters, and Blue Partysupporters—are considered and a pre-determined number of districts areformed. The game proceeds according to alternating, turn-based play;there are 2 players; and moves in categories “E,” “X,” and “R” areallowed.

Game Summary

FIGS. 71-74 illustrate aspects of a strategy game 1500. In this game,two players representing opposing political parties (Red and Blue) viefor political control of region 1502 (e.g., Michigan) by competitivelycreating 14 U.S. congressional districts out of 108 sectors 1504 thatlargely coincide with counties that currently exist in Michigan.

One example of an initial position is illustrated in FIG. 71. At theoutset, the districts are formless and the players know the shape,location, population, and political composition of each sector 1504(i.e. which party its citizens favor and by how much) based on the setof elements 1508 depicted in each sector 1504. During the first phase ofthe game, players gradually build the political districts by assigningsectors to political districts one sector at a time. They may alsoreassign sectors from more populated districts to less populateddistricts in order to better equalize the district populations. Duringthe (optional) second phase of the game, the political margin in eachdistrict is converted into a probability of each party winning thedistrict, and an election is simulated by rolling dice (such as dice1622 of FIG. 44). The winner of the game is the player whose partycontrols more districts than his/her opponent. If both players controlan equal number of districts, the result is a tie. The sets of elementsshown in FIG. 71 are based on the results of the 2016 U.S. presidentialelection, and each includes two numbers. In accordance with the key inTable 1, the first number 1514 (in a rectangle) denotes the sector'svoting population (in thousands). The second number 1516 indicates thesector's voting tendency. FIGS. 72-73 show an example of a first part ofa scoreboard 1510 and a second part of a scoreboard 1512 that may beincluded in a strategy game 1500.

Playing the Game

The rules are very similar to Example 28. The main difference is thatthis game has different population thresholds for moves of type 1 and 2.

All moves made during phase 1 must be of type 1, 1A, 2, 2A, 3, or 3Abelow.

Phase 1 is divided into three stages. During stage 1, only moves of type1, 2, and 3 are allowed. Play enters stage 2 if (i) no more moves oftype 1 exist and (ii) fewer than 14 districts have been established.During stage 2, only moves of type 1A, 2, and 3 are allowed. Play entersstage 3 immediately after the 14^(th) district is established. Duringstage 3, the next move must be of type 2 or 3A if at least one move oftype 2 exists. Otherwise, the next move must be of type 2A or 3A. Thisphase of the game ends when no more legal moves exist.

The six types of legal moves are as follows:

-   -   1 Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (i) The        sector must be at least three steps away from all previously        placed home base markers. (ii) There must be space to grow this        district into a connected district with a population of at least        320.    -   1A Establish a new district by placing its home base marker on a        vacant sector. This move must meet two requirements. (i) The        sector must be in the most populated open space on the board.        (An open space is a connected group of vacant sectors.) (ii)        Among the sectors satisfying the criterion above, the sector        must be the farthest (in number of steps) from a previously        placed home base marker. Ties may be broken arbitrarily.    -   2 Expand a district by placing one of its expansion markers on a        vacant sector. This move must meet four requirements. (i) The        district must remain connected. (ii) The district's initial        population before the move is 320 or less. (iii) The district's        new population—including the new sector and any sectors that are        captured—is 340 or less. (iv) No district may be trapped. A        district is trapped if (a) it, and the open spaces beside it, is        surrounded either by the edge of the board on one side and a        single district on the other side or by a single district on all        sides and (b) its population plus the populations of all open        spaces beside it is 320 or less.    -   2A Expand the least populous expandable district by placing one        of its expansion markers on a vacant sector. The district must        remain connected and no sectors may be captured. (A district is        expandable if there is at least one vacant sector adjacent to        it.)    -   3 (Same as in Example 28)    -   3A (Same as in Example 28)

Example of Play

We now provide an example of play. After 127 moves have been made—64 bythe player representing the Red Party and 63 by the player representingthe Blue Party—a final position as shown in FIG. 74 may be reached, withhome base markers 1518 and expansion markers 1520 placed in a mannerthat defines 14 districts within the region 1502. No legal moves existin this board position, so phase 1 of play concludes. As expected, thedistrict populations sum to 4548, and the population of the averagedistrict is 324.857=4548/14. However, only one district—Purple—has apopulation equaling 324 or 325.

Political District Population Margin Brown 269 +65 Blue Red 348 +52 RedOrange 330 +21 Red Yellow 331 +49 Red Lt. Green 345 +70 Red Dk. Green328 +14 Red Lt. Blue 350  +92 Blue Dk. Blue 272  +24 Blue Purple 324 +21 Blue Pink 338 +34 Red Gray 356   +2 Blue Black 339  +2 Red White338 +81 Red Gold 280 +108 Blue  Overall 4548 11

If phase 2 is not played, the game ends, and the Red Party wins by ascore of 8 districts to 6 districts.

Example 30

Strategy games of this example 30 have taxonomic codeA/G/C?/N/3/?/U/2-3/EXRBF. They are analog, multi-player, nonpoliticalgames with turn-based play and complex two-dimensional sectors that aredesigned for 2-3 players. Each player represents a tribe. The playersare provided with a playing surface illustrating a region divided into anumber of sectors. The premise is that three tribes have been fightingwars against each other within the region illustrated on the board formore than a century. After significant bloodshed and no clear winner,they have decided to peacefully settle their differences by formingdistricts that various tribes will inhabit upon conclusion of the game.

During the game, the players organize the region into districts suchthat (i) each sector is assigned in its entirety to exactly one districtand (ii) each district is a single connected piece.

At the outset the districts are formless and the players are informed ofeach sector's precise shape, location, and set of elements. The set ofelements for each sector provides the intensity of each of threeelements—rivers, plants, and mammals—in the sector.

During play, the players take alternating turns, with each player takinga single turn before any other player takes another turn. During aplayer's turn, the player must make one of the following moves: (E)establish a new district by assigning a first sector to it; (X) expandan already established district by assigning a new, previouslyunassigned sector to the district; (R) reassign a sector from oneestablished district to another established district; (B) break up twoadjacent districts by returning all sectors assigned them to unassignedstatus; and (F) freeze a given district so that no player may modify thedistrict during the next 8 turns. Each player may make each move B and Fat most once during the game. Each player has no limit on the number ofmoves E, X, and R that he/she plays. Each district must be connected atall times during play. In general, any player may use their move tocontribute to the construction, destruction, or freezing of any districtduring any of his/her turns. Play ends when no legal moves exist.

The method of scoring at the end is nontrivial and relates to thesuitability (i.e. habitability) of each district for each tribe.

Tribes A, B, and C have different habitability criteria. Members ofTribe A depend on fishing for sustenance and are allergic to plants.That is, Tribe A considers rivers as a resource and plants as a hazard,and it is indifferent to mammals. Members of Tribe B depend onplants/farming for sustenance and are allergic to mammals. That is,Tribe B considers plants as a resource and mammals as a hazard, and itis indifferent to rivers. Members of Tribe C depend on hunting forsustenance and are very poor swimmers. That is, Tribe C considersmammals as a resource and rivers as a hazard, and it is indifferent toplants.

Once the districts are finalized (i.e., when no legal moves exist), theintensity of each element in each district is computed by summing theintensities of the element in the sectors comprising the district.

At the end of the game, each tribe receives points for each district asfollows.

-   -   If the total intensity of rivers exceeds the total intensity of        plants in a district, Tribe A considers the district habitable        and receives [(river intensity)−(plant intensity)] points for        that district. Otherwise, Tribe A receives zero points for that        district.    -   If the total intensity of plants exceeds the total intensity of        mammals in a district, Tribe B considers the district habitable        and receives [(plant intensity)−(mammal intensity)] points for        that district. Otherwise, Tribe B receives zero points for that        district.    -   If the total intensity of mammals exceeds the total intensity of        rivers in a district, Tribe C considers the district habitable        and receives [(mammal intensity)−(river intensity)] points for        that district. Otherwise, Tribe C receives zero points for that        district.

Each tribe's point total at the end of the game equals the sum of thepoints it receives in all districts. The winner is the player (i.e.tribe) with the most points at the end of the game.

Example 31

Strategy games of this example 31 have taxonomic codeA/G/C?/N/6/?/U/2-6/EXRBF. They are analog, multi-player, nonpoliticalgames with turn-based play and complex two-dimensional sectors that aredesigned for 2-6 players. Each player represents an interplanetarytransportation company. In one example, the premise is that sixinterplanetary transportation companies dominate the economy of theregion of the Milky Way Galaxy in the year 2388. After decades of chaosin the transportation market, the companies have decided, for theirmutual benefit, to set standard transportation rates within the galaxyby dividing it into districts. After the districts are formed, directtransportation between planets will only take place (i) within districtsand (ii) between adjacent districts. No other direct transportationservices will be offered.

During the game, players organize the galaxy—which is already dividedinto C sectors that (i) may not be further divided, (ii) do not overlap,and (iii) together cover the entire galaxy—into a given number, D, ofdistricts (where 2≤D<C) such that (a) each sector belongs in itsentirety to exactly one district and (b) each district is comprised of aset of adjacent sectors.

At the outset the districts are formless and the players are informed ofeach sector's precise shape, location, and set of elements, which inthis case is a set of planets. The set of planets in a sector consistsof six numbers which respectively represent the number of each planettype—agricultural, metropolitan, scholarly, industrial, medical, andecological—in the sector.

Players take turns in rotating fashion. During a player's turn, theplayer must make one of the following moves: (E) establish a newdistrict by assigning a first sector to it; (X) expand an alreadyestablished district by assigning a new, previously unassigned sector tothe district; (R) reassign a sector from one established district toanother established district; (B) break up two or three adjacentdistricts by returning all sectors previously assigned them tounassigned status; and (F) freeze a given district so that no player maymodify the district during the next 5 turns. Each player may make eachmove B and F at most once during the game. Each player has no limit onthe number of moves E, X, and R that he/she plays. Each district must beconnected at all times during play. In general, any player may use theirmove to contribute to the construction, destruction, or freezing of anydistrict during any of his/her turns. Play ends when no legal movesexist.

The method of scoring is nontrivial; it relates to the profitability ofeach district and each pair of adjacent districts for each company.

Each company specializes in a different kind of transportation andtherefore has a different perspective on profitability. Company Aspecializes in transporting food and food equipment between agriculturaland metropolitan planets. Company B specializes in transporting studentsand researchers between metropolitan and scholarly planets. Company Cspecializes in transporting workers and engineers between scholarly andindustrial planets. Company D specializes in transporting injuredworkers between industrial and medical planets. Company E specializes intransporting people and medicinal plants between medical and ecologicalplanets. Company F specializes in transporting flora and fauna betweenecological and agricultural planets.

Once the districts are finalized (i.e., once no legal moves exist), thetotal number of each planet type in each district is computed by summingthe number of that planet type in the sectors comprising the district.Each company receives points for each district as follows.

-   -   Company A receives 3 points for each transportation leg that it        can service in the district, i.e. 3 points for each unique pair        of planets—one agricultural and one metropolitan—in the        district. So Company A gets [3*(no. of agricultural planets in        district)*(no. of metropolitan planets in district)] points for        the district.    -   Company B receives 3 points for each transportation leg that it        can service in the district, i.e. 3 points for each unique pair        of planets—one metropolitan and one scholarly—in the district.        So Company B gets [3*(no. of metropolitan planets in        district)*(no. of scholarly planets in district)] points for the        district.    -   Company C receives 3 points for each transportation leg that it        can service in the district, i.e. 3 points for each unique pair        of planets—one scholarly and one industrial—in the district. So        Company C gets [3*(no. of scholarly planets in district)*(no. of        industrial planets in district)] points for the district.    -   Company D receives 3 points for each transportation leg that it        can service in the district, i.e. 3 points for each unique pair        of planets—one industrial and one medical—in the district. So        Company D gets [3*(no. of industrial planets in district)*(no.        of medical planets in district)] points for the district.    -   Company E receives 3 points for each transportation leg that it        can service in the district, i.e. 3 points for each unique pair        of planets—one medical and one ecological—in the district. So        Company E gets [3*(no. of medical planets in district)*(no. of        ecological planets in district)] points for the district.    -   Company F receives 3 points for each transportation leg that it        can service in the district, i.e. 3 points for each unique pair        of planets—one ecological and one agricultural—in the district.        So Company F gets [3*(no. of ecological planets in        district)*(no. of agricultural planets in district)] points for        the district.

Each company also receives points for each pair of adjacent districts(e.g. X and Y) as follows.

-   -   Company A gets 1 point for each transportation leg that it can        service between Districts X and Y. So Company A gets [(no. of        agricultural planets in X)*(no. of metropolitan planets in        Y)]+[(no. of agricultural planets in Y)*(no. of metropolitan        planets in X)] points for this pair of districts.    -   Company B gets 1 point for each transportation leg that it can        service between Districts X and Y. So Company B gets [(no. of        metropolitan planets in X)*(no. of scholarly planets in        Y)]+[(no. of metropolitan planets in Y)*(no. of scholarly        planets in X)] points for this pair of districts.    -   Company C gets 1 point for each transportation leg that it can        service between Districts X and Y. So Company C gets [(no. of        scholarly planets in X)*(no. of industrial planets in Y)]+[(no.        of scholarly planets in Y)*(no. of industrial planets in X)]        points for this pair of districts.    -   Company D gets 1 point for each transportation leg that it can        service between Districts X and Y. So Company D gets [(no. of        industrial planets in X)*(no. of medical planets in Y)]+[(no. of        industrial planets in Y)*(no. of medical planets in X)] points        for this pair of districts.    -   Company E gets 1 point for each transportation leg that it can        service between Districts X and Y. So Company E gets [(no. of        medical planets in X)*(no. of ecological planets in Y)]+[(no. of        medical planets in Y)*(no. of ecological planets in X)] points        for this pair of districts.    -   Company F gets 1 point for each transportation leg that it can        service between Districts X and Y. So Company F gets [(no. of        ecological planets in X)*(no. of agricultural planets in        Y)]+[(no. of ecological planets in Y)*(no. of agricultural        planets in X)] points for this pair of districts.

Each company's point total at the end of the game equals the sum of thepoints it receives in all districts plus the sum of the points itreceives in all pairs of adjacent districts. The winner is the player(i.e. company) with the most points at the end of the game.

From the foregoing, it will be appreciated that although specificexamples have been described herein for purposes of illustration,various modifications may be made without deviating from the spirit orscope of this disclosure. For example, even though the Examples aredescribed as being analog, they could alternatively be digital, and anycomponent of the games could be digitally represented. It is thereforeintended that the foregoing detailed description be regarded asillustrative rather than limiting, and that it be understood that it isthe following claims, including all equivalents, that are intended toparticularly point out and distinctly claim the claimed subject matter.

What is claimed is:
 1. A strategy game for at least one playercomprising: a region comprising a bounded shape having an area dividedinto a plurality of sectors, wherein each sector comprises a boundedshape having an area within the region that does not overlap with anyother sector, and each sector contains a set of elements, each elementof the set of elements having a type and quantity; wherein, during play,each player makes one move per turn, according to a set of rulesdefining types of moves that can be made by the at least one player andrestrictions governing how districts can be formed from the plurality ofsectors, in pursuit of combining the plurality of sectors into a givennumber of districts in a manner that seeks to achieve a pre-defined goalbased on an aggregation of the elements within each district.
 2. Thestrategy game of claim 1, wherein each element of the set of elements isselected from the group consisting of: resources, hazards, and scraps.3. The strategy game of claim 1, wherein the pre-defined goal isselected from the group consisting of: maximizing a portion of the givennumber of districts that contain at least a certain level of at leastone of the elements of the set of elements; minimizing a portion of thegiven number of districts that contain at least a certain level of atleast one of the elements of the set of elements; and maximizing anumber of points earned by at least one of the players, where the numberof points earned by the at least one player depends on the aggregationof the elements within each district.
 4. The strategy game of claim 1,wherein the strategy game further includes at least two politicalparties, and the set of elements in each sector represents a margin ofvoters within the sector that support each political party.
 5. Thestrategy game of claim 4, wherein the pre-defined goal is selected fromthe group consisting of: maximizing a portion of the given number ofdistricts controlled by one of the political parties, and equalizing aportion of the given number of districts controlled by each of thepolitical parties.
 6. The strategy game of claim 1, further comprising aset of markers, wherein the set of markers comprises a plurality ofmarker subsets, each marker subset representing a district; wherein eachmarker subset comprises a plurality of markers, and each marker withinthe marker subset is configured to be placed on a sector to assign thesector as being part of the district represented by the marker subset.7. A method of playing a strategy game for at least one playercomprising: providing a region comprising a bounded shape having an areadivided into a plurality of sectors, wherein each sector comprises abounded shape having an area within the region that does not overlapwith any other sector, and each sector contains a set of elements, eachelement of the set of elements having a type and quantity; making onemove per turn per player, according to a set of rules defining types ofmoves that can be made by the at least one player and restrictionsgoverning how districts can be formed from the plurality of sectors, inpursuit of combining the plurality of sectors into a given number ofdistricts in a manner that seeks to achieve a pre-defined goal based onan aggregation of the elements within each district.
 8. The method ofclaim 7, further comprising: providing a set of markers, wherein the setof markers comprises a plurality of marker subsets, each marker subsetrepresenting a district; wherein each marker subset comprises aplurality of markers, and each marker within the marker subset isconfigured to be placed on a sector to assign the sector as being partof the district represented by the marker subset.
 9. The method of claim8, wherein the types of moves that can be made by the at least oneplayer include assigning a sector to a district, and the assigningcomprises placing a marker from one of the plurality of marker subsetsonto the sector.
 10. The method of claim 7, wherein the types of movesthat can be made by the at least one player includes establishing a newdistrict by assigning a first sector to a district to create anestablished district.
 11. The method of claim 7, wherein the types ofmoves that can be made by the at least one player includes expanding adistrict by assigning an unassigned sector to a previously establisheddistrict.
 12. The method of claim 7, wherein the types of moves that canbe made by the at least one player includes reassigning a sector fromone district to another district.
 13. The method of claim 7, wherein therestrictions governing how any of the districts can be formed from theplurality of sectors include a restriction that each district must becomprised of one or more sectors.
 14. The method of claim 7, wherein therestrictions governing how any of the districts can be formed from theplurality of sectors include a restriction that, at the conclusion ofplay, each district must have a number of sectors selected from thegroup consisting of: a minimum number of sectors, a maximum number ofsectors, or a specific pre-defined number of sectors.
 15. The method ofclaim 7, wherein the restrictions governing how any of the districts canbe formed from the plurality of sectors include a restriction that eachdistrict must be connected.
 16. The method of claim 7, wherein therestrictions governing how any of the districts can be formed from theplurality of sectors include a restriction that, at the end of play,every sector within the region must be assigned to a district.
 17. Themethod of claim 7, wherein the restrictions governing how any of thedistricts can be formed from the plurality of sectors include arestriction that each sector is assigned in its entirety to exactly onedistrict.
 18. The method of claim 7, wherein the step of making one moveper turn per player comprises alternating moves between each player. 19.The method of claim 7, wherein the step of providing a region comprisesproviding a separate identical region to each player, and the step ofmaking one move per turn per player comprises each player independentlyassigning a sector to a district within the region provided to thatplayer.
 20. The method of claim 7, wherein the step of making one moveper turn comprises one of: (a) establishing a new district by assigningthe first sector that belongs to it; (b) expanding a district byassigning an unassigned sector to an already established district; (c)reassign a sector from one district to another district, (d) breaking upone or more adjacent districts by un-assigning all sectors that belongto these adjacent districts; and (e) freezing a district by prohibitingany player from modifying the district during the next one or moreturns.